The subject of this paper is the analytic approximation of solution to stochastic differential delay equations with Poisson jump. We introduce approximate methods for stochastic differential equations driven by Poisson random measure, as well as for those driven by Poisson process. In both cases, approximate equations are defined on equidistant partitions of the time interval, and their coefficients are Taylor approximations of the coefficients of the initial equation. It will be shown that the approximate solutions converge in the $L^p$-sense and almost surely to the solutions of the corresponding initial equations. The order of the $L^p$-convergence of the approximate solutions to the solution of the initial equation is established and it increases when the number of degrees in Taylor approximations of coefficients increases.