Incomplete exponential sums over exponential functions

We extend some methods of bounding exponential sums of the type ${\displaystyle \sum _{n\le N}{e}^{2\pi ia{g}^n\mathrm{/}p}}$ to deal with the case when $g$ is not necessarily a primitive root. We also show some recent results of Shkredov concerning additive properties of multiplicative subgroups imply new bounds for the sums under consideration.