Is there sth in cvx that can give an invalid solution priority over another invalid solution ? meaning, which one was closer to find a valid solution ?

If CVX/solver has determined your problem to be infeasible, then it will tell you it is infeasible, and “the ball game is over”, i.e., CVX is done.

If your problem is infeasible, but you want to find a point which is closest to feasibility, then create a new optimization (CVX) problem whose objective function minimizes the infeasibility according to the criterion you want. For instance in the norm you “like”, such as 2 or inf. You can put all the original problem constraints into the new problem objective function and then solve an unconstrained problem, or keep some constraints as constraints (if feasible and desired) and put the other constraints in the objective function.

you mean something like dual variables ?

I mean formulating and solving with CVX a problem to minimize infeasibility (in some sense) of the original problem.