^^^^^^i belive its yes you still got an mot as long as the retest does not find any thing non road worthy
as for fix and take it back the gargages round here say once you take it away you pay for a retest theres no grace as the car has to be cheaked all over as you could have broken some thing that was fine but now is not after the repair
if they do the work its up to the mot tester if he needs to cheak every thing but no retest fee if they do the work