If y\geq 0 then a model for t\geq y\exp(y) is essentially in https://docs.mosek.com/modeling-cookbook/expo.html#lambert-w-function, equation (5.10), up to renaming variables. Throwing in x we get the following model for t\geq y\exp(y/x), x,y\geq 0:
t\geq y\exp(u/y)
ux\geq y^2
The first one is an exponential cone the other is rotated quadratic.
Now in general it should be OK to replace y with a positive z with z\geq |y| but frankly I haven’t checked all the correct monotonicity details. Either it works or I don’t see another way.