For an LP problem, I want to use the last basic variables for resolving the problem aka. warm start or hot-start. Can I pass the required information to the Mosek solver or Gurobi?
Not when using CVX .
And is it possible to add a constraint once the problem is solved, such that when solving it with the new constraint it is way faster ? For example I know that it is possible in JuMP.
After solving a problem with CVX, you are free to solve it again, or solve a related problem, with whatever constraints you want. But CVX and the solver it calls will start from scratch each time.
JuMP has some warm/hot starting capability, as well as some ability to modify problems without starting from scratch. CVX does not. So perhaps you are thinking of a problem in which JuMP is warm or hot started after an additional constraint is added, perhaps rendering that solution non-optimal given the added constraint, then using, for instance, Dual Simplex, to get to optimality more quickly on the new problem.