Are there problems that can be solved in time $2^{n-q^c}$ with $q$ qubits?
I think the scaling $2^{n/q^c}$ is too much to ask for. Even $poly(q) 2^{O(n-q)}$ would represent an exponential speedup for each additional qubit. 

And indeed, such a problem is known: simulating a quantum circuit of $n$ logical qubits on a small hybrid quantum-classical computer with only few (perfect) physical qubits $q\leq n$ has this scaling. See: https://arxiv.org/abs/1506.01396