The mosquito had range, and it had firepower, but it was NOT a fighter plane. It was never used as a pure fighter [EDIT: as in send up a squadron on a fighter sweep over the channel, as in engage only enemy fighters and clear the skies], and for a good reason. It was not highly manuverable. Oh, to be sure, it could manuver well... for a bomber, but then so could the Ju88 and so could the He111. Neither of those were put into long range fighter duties, even though the Ju88 had heavy fighter variants.
The simple matter is that, aside from pure speed, the Mosquitos would have been totally outclassed by the average 109s and 190s.