We introduce a mixed {\em generalized} Dynkin game/stochastic control with
Ef-expectation in a Markovian framework. We study both the case when
the terminal reward function is supposed to be Borelian only and when it is
continuous. We first establish a weak dynamic programming principle by using
some refined results recently provided in \cite{DQS} and some properties of
doubly reflected BSDEs with jumps (DRBSDEs). We then show a stronger dynamic
programming principle in the continuous case, which cannot be derived from the
weak one. In particular, we have to prove that the value function of the
problem is continuous with respect to time t, which requires some technical
tools of stochastic analysis and some new results on DRBSDEs. We finally study
the links between our mixed problem and generalized Hamilton Jacobi Bellman
variational inequalities in both cases