But you can fly- for private recreational flying! What they are saying is 'while you are full-time with us, being paid for a full time job, you don't do paid-for flying employment with anybody else unless we vet it (and guess what the answer will be!).
Companies are now required to monitor 'extraneous' flying by their pilots. All such extraneous flying goes towards your mandatory weekly/monthly/annual flying hour limits. So the company expects to get full contractual flying limits out of you...for the benefit of paying you a full-time salary. Do you think it reasonable to say to them 'well- I've been flying 15 hours this month, so would you reduce my duty hours by 12 flying hours please (but keep the full time salary coming in please!)'. Or 'I can't do the Paphos on Friday because I flew my Cherokee to Plymouth on Monday!- but that's not MY problem!' It is your problem. you are a full time professional pilot. Sadly, private flying WILL affect your job. there is a simple solution- go part-time if you can- that way you can fly as much as you like in your own time for part-time pay, and it won't affect your profession. Exactly how much do you expect from your company- it's perfectly reasonable?