In 1943 a Mosquito FB.Mk VI without the flame dampers would flat out run away from any Bf109 and most Fw190s. Certainly the Bf109G-6 and Fw190A-5 could not catch one in a tail chase.
In 1944 things are quite different. The Bf109G-6 with MW50, Bf109G-14, Bf109G-10, Bf109K-4, Fw190A-8, Fw190A-9, Fw190D-9, ect would all be able to run one down.
I am not sure where the Fw190A-6 stands.
So you cannot categoreically say that the Mossie FB.VI could or could not run. You must give the context of the year.
As to fighting back, I am aware of the results of one Fw190 squad vs Mosquito FB.VI squad combat. Four Mossies were shot down and two Fw190s were shot down.
Removing the dampers would raise the survivability of the Mossie in AH as any additional speed helps, but there are still many fighters that could run one down and those fighters tend to be the most popular.