  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 $210\ \mu\mathrm{eV}$
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.
