Is there an equation for efficiency of a 2-radix FFT as you raise $k$? I know the efficiency for an $N$ point 2-radix FFT is $N\log_2(N)$ but assuming $k\leq N$, what if you were looking for the efficiency of calculating $k$ positions of the FFT? Would the efficiency be $k\log_2(N)$?
No.

The FFT is pretty much an "all or nothing" algorithm. There is almost no efficiency gain in reducing the number of desired outputs. 

For $k \ll N$ the Goertzel algorithm is typically the better choice. See https://en.wikipedia.org/wiki/Goertzel_algorithm. The break even point is quite small. Maybe $k = 4\cdot log_2(N)$ or thereabouts.