Jesus Christ had existed before His incarnation as the God that men had seen in the O.T. Jesus testified that Moses had written of Him.
John 5:39 Search the scriptures; for in them ye think ye have eternal life: and they are they which testify of me. 40 And ye will not come to me, that ye might have life...…….46 For had ye believed Moses, ye would have believed me; for he wrote of me. 47 But if ye believe not his writings, how shall ye believe my words?
If you are wondering about 1 Timothy 2:5 as signifying the man Christ Jesus, I believe that is referring to the Son that had become incarnated, crucified, resurrected, and ascended to the right hand of God the Father to be our only Mediator between us and God the Father. It is an important significant role so that every believer knows that there is no other Divine Mediator between us and the Father to go to; Not Mary, not the departed saints, and not even the Holy Spirit as Jesus explains why.
John 14:6 Jesus saith unto him, I am the way, the truth, and the life: no man cometh unto the Father, but by me.....13 And whatsoever ye shall ask in my name, that will I do, that the Father may be glorified in the Son. 14 If ye shall ask any thing in my name, I will do it.
Matthew 28:18 And Jesus came and spake unto them, saying, All power is given unto me in heaven and in earth. 19 Go ye therefore, and teach all nations, baptizing them in the name of the Father, and of the Son, and of the Holy Ghost: 20 Teaching them to observe all things whatsoever I have commanded you: and, lo, I am with you always, even unto the end of the world. Amen.
Does the Holy Spirit has a hand in answering prayers; yes, but not in the way we would think when He will give the glory of answered prayers to the Son as He is the One that answers prayers so that the Father may be glorified in the Son for answers to prayers.
John 16:13 Howbeit when he, the Spirit of truth, is come, he will guide you into all truth: for he shall not speak of himself; but whatsoever he shall hear, that shall he speak: and he will shew you things to come. 14 He shall glorify me: for he shall receive of mine, and shall shew it unto you. 15 All things that the Father hath are mine: therefore said I, that he shall take of mine, and shall shew it unto you.
How? Jesus takes our intercessions, the silent intercessions of the Holy Spirit's, and His own intercessions to the Father so whenever the Father says "Yes" to any of those intercessions, the Son answers the prayer for why we give the Father "thanks" in Jesus's name.
I am not sure if that is the reason why you referenced the man Christ Jesus, but only God hears prayers and Jesus Christ is the only way to approach God the Father in prayer, BUT only Christ Jesus answers prayers because He is also God so that the Father may be glorified in the Son for answers to prayers. The "man Christ Jesus" is to signify the Son of God/Son of Man and not think the Holy Spirit is Another Mediator between God & man also when there is only One Mediator between us & the Father.