I don’t see how you got your formulation. Are you incorporating other constraints?
Nevertheless, by including the following in your CVX program
variables x y t;
abs(x) <= geo_mean([t,t,y]);
x >= 0
y>=0
you can use t in place of x*sqrt(x/y).
The program
cvx_begin
variables x y t;
minimize(t)
subject to
abs(x) <= geo_mean([t,t,y]);
x >= 0
y>=0
cvx_end
results, within solver tolerance, in optimal value of x and t both equal to zero, and y being any positive number.
Adding the constraint x >= 1 makes the problem ill-posed, because the infimum of 0 is achieved (actually, not achieved) in the limit as y \rightarrow \infty. Solvers don’t like that.