Large coupling strengths in exciton-photon interactions are important for quantum photonic network, while strong cavity-quantum-dot interactions have been focused on s-shell excitons with small coupling strengths. Here we demonstrate strong interactions between cavities and p-shell excitons with a great enhancement by the textit{in situ} wave-function control. The p-shell excitons are demonstrated with much larger wave-function extents and nonlocal interactions beyond the dipole approximation. Then the interaction is tuned from the nonlocal to local regime by the wave-function shrinking, during which the enhancement is obtained. A large coupling strength of $210 mumathrm{eV}$ has been achieved, indicating the great potential of p-shell excitons for coherent information exchange. Furthermore, we propose a distributed delay model to quantitatively explain the coupling strength variation, revealing the intertwining of excitons and photons beyond the dipole approximation.
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