$P$--$V$ criticality, Joule--Thomson expansion, and holographic heat engine of charged Hayward-AdS black holes with a cloud of strings and perfect fluid dark matter

We construct the charged Hayward-anti-de Sitter (AdS) black hole (BH) with a cloud of strings (CS) and perfect fluid dark matter (PFDM), and analyze its extended thermodynamic phase structure. The Hayward parameter $g$ replaces the central singularity with a de Sitter (dS) core, while the CS parameter $a$ and the PFDM parameter $\beta$ encode astrophysically motivated matter content. Treating the cosmological constant as pressure, we derive the thermodynamic quantities, verify the Smarr relation, and establish $P$–$V$ criticality with a van der Waals (vdW)-like small-large BH phase transition and mean-field critical exponents. The Gibbs free energy (GFE) exhibits the characteristic swallowtail below the critical pressure. The Joule-Thomson (JT) expansion yields $T_i^{\mathrm{m}\mathrm{i}\mathrm{n}}\mathrm{/}{T}_c\approx 0.247$, roughly half the Reissner–Nordström-AdS value. The parameters $g$ and $Q$ contract the cooling region, $\beta$ expands it, and $a$ reshapes it non-monotonically. A holographic heat engine with a rectangular cycle gives efficiencies $\eta =0.362$–$0.396$ and Carnot benchmarking ratios $\eta \mathrm{/}{\eta }_C=0.625$–$0.791$ across six configurations. The CS parameter improves the engine efficiency by reducing the enthalpy at fixed thermodynamic volume, while the PFDM parameter degrades it by adding gravitational enthalpy without contributing to the mechanical work.