First of all, please read this FAQ). In particular, read the section “Yes, I am sure my model is convex!”. It is important for you to understand that CVX can’t just be told an expression is convex. It doesn’t work that way.
Now, that said: if t is a positive constant, then
$$\log(1/t+\exp(y))=\log(\exp(-\log t)+\exp(y))$$
and the log_sum_exp function will serve you well here:
log_sum_exp([-log(t),y]) <= x
There is experimental support for the solver SCS in CVX 3.0 beta. This solver can indeed handle exp/log natively. But you still must express your problem in the same way as before; i.e., with log_sum_exp.