Can a preacher direct non-believers to pray a prayer or follow his prayer of repentance and this lead to their salvation?

There are no specific words or phrases that can be recited or prayed that imparts eternal life to a sinner. Yes, people do pray and can even repeat someone else’s prayer when they get saved but it is not the prayer that saves them but rather the point in which they exercise faith in the Gospel of Jesus Christ (1 Corinthians 15:1-4 [1] Moreover, brethren, I declare unto you the gospel which I preached unto you, which also ye have received, and wherein ye stand; [2] By which also ye are saved, if ye keep in memory what I preached unto you, unless ye have believed in vain. [3] For I delivered unto you first of all that which I also received, how that Christ died for our sins according to the scriptures; [4] And that he was buried, and that he rose again the third day according to the scriptures:
See All..., Jesus death, burial and resurrection) and it is at that moment that they become born again. Jesus tells us that simply saying a prayer for the sake of repeating the words was vain (worthless) repetition (Matthew 6:7But when ye pray, use not vain repetitions, as the heathen do: for they think that they shall be heard for their much speaking.
See All..., Luke 18:10-14 [10] Two men went up into the temple to pray; the one a Pharisee, and the other a publican. [11] The Pharisee stood and prayed thus with himself, God, I thank thee, that I am not as other men are, extortioners, unjust, adulterers, or even as this publican. [12] I fast twice in the week, I give tithes of all that I possess. [13] And the publican, standing afar off, would not lift up so much as his eyes unto heaven, but smote upon his breast, saying, God be merciful to me a sinner. [14] I tell you, this man went down to his house justified rather than the other: for every one that exalteth himself shall be abased; and he that humbleth himself shall be exalted.
See All...).

Romans 10:13For whosoever shall call upon the name of the Lord shall be saved.
See All... states that, “whosoever shall call upon the name of the Lord shall be saved,” but that verse is predicated on the heart felt act of faith which in turn brings about the new birth. Notice from the verses that proceed Romans 10:13For whosoever shall call upon the name of the Lord shall be saved.
See All... how that the prayer is one of sincere belief and faith that has been expressed within the person’s heart.

Romans 10:9-12 [9] That if thou shalt confess with thy mouth the Lord Jesus, and shalt believe in thine heart that God hath raised him from the dead, thou shalt be saved. [10] For with the heart man believeth unto righteousness; and with the mouth confession is made unto salvation. [11] For the scripture saith, Whosoever believeth on him shall not be ashamed. [12] For there is no difference between the Jew and the Greek: for the same Lord over all is rich unto all that call upon him.
See All... That if thou shalt confess with thy mouth the Lord Jesus, and shalt believe in thine heart that God hath raised him from the dead, thou shalt be saved. For with the heart man believeth unto righteousness; and with the mouth confession is made unto salvation. For the scripture saith, Whosoever believeth on him shall not be ashamed. For there is no difference between the Jew and the Greek: for the same Lord over all is rich unto all that call upon him.

Since the mouth is connected to expressing the desires of the heart (Matthew 12:34O generation of vipers, how can ye, being evil, speak good things? for out of the abundance of the heart the mouth speaketh.
See All...b, “for out of the abundance of the heart the mouth speaketh.”) then it should not be any surprise as to why confession then is mentioned within the text. Again, notice how in the verses following Romans 10:13For whosoever shall call upon the name of the Lord shall be saved.
See All... the prayer is interwoven with the faith that has been exercised by the sinner.

Romans 10:14How then shall they call on him in whom they have not believed? and how shall they believe in him of whom they have not heard? and how shall they hear without a preacher?
See All... How then shall they call on him in whom they have not believed? and how shall they believe in him of whom they have not heard?

Paul again reiterated this fact to the Philippian jailor when he inquired how he might be saved:

Acts 16:29-31 [29] Then he called for a light, and sprang in, and came trembling, and fell down before Paul and Silas, [30] And brought them out, and said, Sirs, what must I do to be saved? [31] And they said, Believe on the Lord Jesus Christ, and thou shalt be saved, and thy house.
See All... Then he called for a light, and sprang in, and came trembling, and fell down before Paul and Silas, And brought them out, and said, Sirs, what must I do to be saved? And they said, Believe on the Lord Jesus Christ, and thou shalt be saved, and thy house.

When Philip was dealing with the Ethiopian eunuch about salvation he wouldn’t allow him to be baptized until after he had professed faith in Jesus Christ;

Acts 8:36-38 [36] And as they went on their way, they came unto a certain water: and the eunuch said, See, here is water; what doth hinder me to be baptized? [37] And Philip said, If thou believest with all thine heart, thou mayest. And he answered and said, I believe that Jesus Christ is the Son of God. [38] And he commanded the chariot to stand still: and they went down both into the water, both Philip and the eunuch; and he baptized him.
See All...  And as they went on their way, they came unto a certain water: and the eunuch said, See, here is water; what doth hinder me to be baptized? And Philip said, If thou believest with all thine heart, thou mayest. And he answered and said, I believe that Jesus Christ is the Son of God. And he commanded the chariot to stand still: and they went down both into the water, both Philip and the eunuch; and he baptized him.

The Lord himself stated on several occasions how that belief in Him is what grants eternal life:

John 3:14-16 [14] And as Moses lifted up the serpent in the wilderness, even so must the Son of man be lifted up: [15] That whosoever believeth in him should not perish, but have eternal life. [16] For God so loved the world, that he gave his only begotten Son, that whosoever believeth in him should not perish, but have everlasting life.
See All...  And as Moses lifted up the serpent in the wilderness, even so must the Son of man be lifted up: That whosoever believeth in him should not perish, but have eternal life. For God so loved the world, that he gave his only begotten Son, that whosoever believeth in him should not perish, but have everlasting life.

John 3:17-18 [17] For God sent not his Son into the world to condemn the world; but that the world through him might be saved. [18] He that believeth on him is not condemned: but he that believeth not is condemned already, because he hath not believed in the name of the only begotten Son of God.
See All...  For God sent not his Son into the world to condemn the world; but that the world through him might be saved. He that believeth on him is not condemned: but he that believeth not is condemned already, because he hath not believed in the name of the only begotten Son of God.

John 3:36He that believeth on the Son hath everlasting life: and he that believeth not the Son shall not see life; but the wrath of God abideth on him.
See All...  He that believeth on the Son hath everlasting life: and he that believeth not the Son shall not see life; but the wrath of God abideth on him.

In closing, it cannot be emphasized enough when dealing with the unbeliever that they understand that salvation is in the finished work of Jesus Christ and not a specific prayer that they say.

1 John 5:11-12 [11] He answered them, He that made me whole, the same said unto me, Take up thy bed, and walk. [12] Then asked they him, What man is that which said unto thee, Take up thy bed, and walk?
See All...  And this is the record, that God hath given to us eternal life, and this life is in his Son. He that hath the Son hath life; and he that hath not the Son of God hath not life.