  We consider a class of general SDEs with a jump integral term driven by a
time-inhomogeneous random Poisson measure. We propose a two-parameters
Euler-type scheme for this SDE class and prove an optimal rate for the strong
convergence with respect to the $L^p(\Omega)$-norm and for the weak
convergence. One of the primary issues to address in this context is the
approximation of the noise structure when it can no longer be expressed as the
increment of random variables. We extend the Asmussen-Rosinski approach to the
case of a fully dependent jump coefficient and time-dependent Poisson
compensation, handling contribution of jumps smaller than $\varepsilon$ with an
appropriate Gaussian substitute and exact simulation for the large jumps
contribution. For any $p \geq 2$, under hypotheses required to control the
$L^p$-moments of the process, we obtain a strong convergence rate of order
$1/p$. Under standard regularity hypotheses on the coefficients, we obtain a
weak convergence rate of $1/n+\epsilon^{3-\beta}$, where $\beta$ is the
Blumenthal-Getoor index of the underlying L\'evy measure. We compare this
scheme with the Rubenthaler's approach where the jumps smaller than
$\varepsilon$ are neglected, providing strong and weak rates of convergence in
that case too. The theoretical rates are confirmed by numerical experiments
afterwards. We apply this model class for some anomalous diffusion model
related to the dynamics of rigid fibres in turbulence.
