One $O(n^k)$ algorithm requiring only one $O(2^n)$ computation (for all n instances) is P or NP
If the "thing" is polynomial-size, then it's in P/poly, a class that has been studied intensively, although this is a very interesting subclass of it which I don't believe been studied. If you have an algorithm that falls into this subclass, I believe theoretical computer scientists might be interested in it.

If the "thing" can be exponential size, then the "thing" could just be a table of all the answers for the instances of size $n$, so all algorithms in EXPTIME would fall into this class. And in fact, this class would be equal to EXPTIME, since the algorithm R followed by A is in EXPTIME.