for example, -log(1+1/x +1/y) in CVX. I want to solve it, x and y are optimizing variables.

You can easily model

log(1+1/x) <=t

using the trick shown at the third line under entropy

https://docs.mosek.com/cheatsheets/conic.pdf

I think that idea can be extended to

log(1+1/x+1/y) <=t

Two approaches are shown in section 5.2.7 Log-sum-inv of https://docs.mosek.com/modeling-cookbook/expo.html#exponential-cone