  In this paper, we study the numerical approximation of a general second order
semilinear stochastic partial differential equation (SPDE) driven by a additive
fractional Brownian motion (fBm) with Hurst parameter $H>\frac 12$ and Poisson
random measure, more realistic in modelling real world phenomena. To the best
of our knowledge, numerical schemes for such SPDE have been lacked in
scientific literature. The approximation is done with the standard finite
element method in space and three Euler-type timestepping methods in time, more
precisely
  linear implicit method, exponential integrator and exponential Rosenbrock
scheme are used for time discretisation.
  In contract to the current literature in the field for SPDE driven only by
fBm, our linear operator is not necessary self-adjoint and optimal strong
convergence rates have been achieved for SPDE driven only by fBm and SPDE
driven by fBm and Poisson measure.
  The results examine how the convergence orders depend on the regularity of
the noise and the initial data and reveal that the full discretization attains
an optimal convergence rate of order $\mathcal{O}(h^2+\Delta t)$ for the
exponential integrator and implicit schemes (linear operator $A$ self-adjoint
for implicit).
  Numerical experiments are provided to illustrate our theoretical results for
the case of SPDE driven with fBm noise.
