To understand what "let us go on to perfection" refers to, this crucial phrase has to be considered: "... this we will do if God permits".
The point to keep in mind is that this letter writer understands that he is being led by the Holy Spirit.
He shows this clearly in chapter 10 verse 15, saying, "... the Holy Spirit also witnesses to us ..." (present tense).
He is speaking like a teacher who has a sequence of topics in his mind that his readers need to hear. And the next subject to go on to teach "perfection".
Because he is sensitive to the Holy Spirit he says what a Believer today might say "God willing". He says, "... if God permits".
He does not take the Spirit's leading for granted. God may want to him to teach something else and leave "perfection".
This writer's way of Holy Spirit-led teaching (of what Christ is in comparison to the Law) began at verse 1 of chapter 1 until chapter 5 verse 6. There he began to teach about Melchizedek.
But he had to stop (at chapter 5 verse 11) for a Spirit-led interruption in order to exhort, rebuke and encourage these Believers. It continues to chapter 6 verse 3.
"... you have become dull of hearing.
For though by this time you ought to be teachers, you need someone to teach you again the first principles of the oracles of God; and you have come to need milk and not solid food.
For everyone who partakes only of milk is unskilled in the word of righteousness, for he is a babe.
But solid food belongs to those who are of full age, that is, those who by reason of use have their senses exercised to discern both good and evil".
He continued by adding: "Therefore,
leaving the discussion of the elementary principles of Christ, let us go on to perfection, not laying again the foundation of repentance from dead works and of faith toward God,
of the doctrine of immersions, of laying on of hands, of resurrection of the dead, and of eternal judgment.
And this we will do if God permits".
This last remark reveals that he did not want to simply push on just because "perfection" was next in order. Hence, he says, "if God permits".
Permission granted! God did permit him. The teaching about Melchizedek that he had begun (at 5'6) and broke off (at 5'11) he restarted (at 6'20).
At chapter 7 verse 11 he goes on to teaching perfection in chapter 7'11. There he says:
"... if perfection were through the Levitical priesthood
(for under it the people received the law),
what further need was there that another priest should rise according to the order of Melchizedek,
and not be called according to the order of Aaron?".
The law was limited. Most importantly, that sacrifices made by the High Priest could not achieve perfection for anyone.
"... for the law made nothing perfect; on the other hand, there is the bringing in of a better hope, through which we draw near to God" (7'19).