On this specific point, No is the answer. The only deals the PL has the power to assess and re-value if necessary are the deals between a club and an associated party.
(Slight tangent, the associated party rules were substantially introduced because the rules the PL initially adopted didn't cover the relationship between City and Abu Dhabi based companies like Etihad and Etisalat.)
So it is only where the league can say "they are only paying £X because of this relationship between the club and the sponsor." That is where the League can step in and say "the fair market value of the rights this club is selling to that sponsor is £Y, and we will exercise our right to block any deal that is in excess of £Y."
However that right to revalue the contract only arises where it relates to a deal between a club and an associated party. So far as we can tell, there is no association between City and Puma so even if they had been willing to pay City £5bn to be a sleeve sponsor on the away shirt only, the League would not have had the right to re-value that contract.
What we know is that the club has to submit any deal for assessment before they can enter into it, but that does not mean the League go straight to the FMV stage. The assessment process is (a) to determine whether the sponsor is a related party and (b) if so, only then to determine what the fair market value of the transaction would be. As far as we know, the Puma deal has gone through, so either the League accepted that Puma was not a related party, or accepted that the deal was for a fair market value. I don't know, but I assume the former.