Do I have to notify someone or file a flight plan for a private flight in the UK?
Stapleford and Damyns Hall are both in class G airspace, so there would be no need to file a flight plan or ask permission of anyone to make the flight. The only thing you might need to do is call Damyns Hall in case they are PPR (Prior Permission Required). VFR flights generally don't need any sort of flight plan unless you are going abroad or to the channel islands. If you are going a long distance over water it's advisable as well. If flying into Damyns Hall keep in mind that the London City zone is very close by, so make sure you stay clear of it. The London TMA is at 2,500ft as well. 

Both airfields are not controlled, however both have AG radios who try to coordinate flights, you technically don't have to communicate with them but you might not be allowed to use those airfields in the future if you don't use it.