When we apply a magnetic field on a moving charge, under Lorentz force, the charge changes its trajectory and moves in a circle. However, as the charge circles around, it creates around itself a magnetic field (Biot-Savart law). As a result, these two fields add up vectorially according to superposition. My question is: is my description above correct? Would the motion be rather complicated then?
Yes your arguments are valid, but in the standard treatment of problems in classical electrodynamics we neglect the size of the magnetic field induced by the moving charge in comparison to the external field. This is why literature sometimes calls these charges 'test charges', because they only test the external field, while not being field inducing themselves. Of course, if you have a significant amount of charges in an external field, you will have to consider this effect of field induction (see i.e. plasma physics). It is a matter of perturbation theory, to get the best approximation that fits your needs (or the precision of your experiment).