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The new results on the muon anomalous magnetic moment (AMM) published by Fermilab in 2021, did not lead to a reduction of its long-pending deviation from the Standard Model (SM) value by more than 4$σ$. The explanation of this discrepancy by adding new particles to the theory puts many new physics models under tension when combined with the null results of the LHC direct searches for new particles. In this paper, we investigate the CP-violating Next-to-Minimal Supersymmetric extension of the SM (NMSSM) with and without an inverse seesaw mechanism. We compute the one-loop supersymmetric contributions to the AMM and the two-loop Barr-Zee-type diagrams with effective Higgs couplings to photons for the leptonic electric dipole moments (EDMs). The effects of the extended (s)neutrino sector on the muon AMM and on the mass of the SM-like Higgs boson can be significant. Complex phases can have an important impact on the AMM. On the other hand, the stringent limits from the EDMs on the complex phases have to be taken into account. Our calculations have been implemented in the Fortran codes NMSSMCALC and NMSSMCALC-nuSS which are publicly available. Besides the leptonic AMMs and EDMs, these programs can compute the Higgs boson masses and mixings, together with Higgs boson decay widths and branching ratios taking into account the mostup-to-date higher-order corrections in the NMSSM with and without inverse seesaw mechanism.