Is there a specific CVX function to define a power cone constraint or an exponential one?
Power cone is not directly supported by CVX, even though it is by Mosek. So you will have to handle it using what CVX does offer, perhaps some combination of (rotated) second order cones and geometric mean - there are some examples scattered on this forum, and you can look at https://docs.mosek.com/modeling-cookbook/powo.html for hints, or section 3.2.6 of version 2 (not later) of the Mosek Modeling Cookbook, available at https://docs.mosek.com/MOSEKModelingCookbook-v2.pdf .
CVX does support exponential cone constraints.
{x,y,z} == exponential(1)
constrains y > 0, y*exp(x/y) <= z