Effect of dark energy on photon orbits and thermodynamic phase transition for Hayward anti-de Sitter black holes

We investigate a black hole solution analogous to Hayward regular black holes with a negative cosmological constant surrounded by quintessence — the Hayward–Kiselev anti-de Sitter (AdS) black hole, derived from Einstein equations coupled with nonlinear electrodynamics. The solution reveals a singularity due to the quintessential dark energy, and we have outlined the conditions necessary for regularity. When pressure, e.g., $P=0.016$, the effect of quintessential dark energy results in an otherwise absent van der Waal phase transition and second-order phase transition. Interestingly, the impact of dark energy decreases critical temperature ($T_c$), whereas critical pressure ($P_c$) increases. Investigating the correlation between photon sphere radius and the first-order phase transition exhibited non-monotonic behaviour among the photon sphere radius, nonlinear charge parameter, temperature, and pressure under certain conditions. Variations in the photon sphere radius and impact parameter before and after the phase transition serve as order parameters. The critical exponents $\Delta r_{ps}$ and $\Delta u_{ps}$ near the critical point consistently approach 1/2, suggesting that $r_{ps}$ and $u_{ps}$ can be order parameters for black hole phase transitions and indicating a potential universal gravitational relationship near the critical point within a black hole thermodynamic system. We also analysed the previously unexplored Kiselev-AdS black hole, which emerges as a particular case in the limit $g\to 0$.