Exceptional points (EPs) of both eigenvalue and eigenvector degeneracy offer remarkable properties of the non-Hermitian systems based on the Jordanian form of Hamiltonians at EPs. Here we propose the perturbation theory able to underpin the physics in the vicinity of the higher-order EPs. The perturbation theory unveils lifting of degeneracy and origin of the different phases merging at the EP. It allows us to analyze the photonic local density of states and resonance energy transfer, determining their spectral behaviors in a general form. Resonant energy transfer is investigated in analytical and numerical examples. We analytically find the resonance energy transfer rate near the third-order EP occurring in the system of three coupled cavities and reveal singularities caused by the interplay of the perturbation and frequency detuning from degenerate eigenfrequency. Numerical simulation of the coupled-resonator system reveals the vital role of a mirror for switching to the EP of the doubled order and corresponding enhancement of the resonance energy transfer rate. Our investigation sheds light on the behavior of nanophotonic systems in non-Hermitian environments.