We study the near-threshold J/ψ meson photoproduction from protons and nuclei by considering incoherent direct non-resonant (γp→J/ψp, γn→J/ψn) and two-step resonant (γp→Pci+→J/ψp, γn→Pci0→J/ψn, i=1, 2, 3, 4; Pc1+,0=Pc+,0(4312), Pc2+,0=Pc+,0(4337), Pc3+,0=Pc+,0(4440), Pc4+,0=Pc+,0(4457)) charmonium production processes. We calculate the absolute excitation functions, energy and momentum distributions for the non-resonant, resonant and for the combined (non-resonant plus resonant) production of J/ψ mesons on protons as well as, using the nuclear spectral function approach, on carbon and tungsten target nuclei at near-threshold incident photon energies by assuming the spin-parity assignments of the hidden-charm resonances Pc+,0(4312), Pc+,0(4337), Pc+,0(4440) and Pc+,0(4457) as JP=(1/2)−, JP=(1/2)−, JP=(1/2)− and JP=(3/2)− within three different realistic scenarios for the branching ratios of their decays to the J/ψp and J/ψn modes (0.25, 0.5 and 1%). We show that will be very hard to measure the Pci+ pentaquark states through the scan of the J/ψ total photoproduction cross section on a proton target in the near-threshold energy region around the resonant photon energies of 9.44, 9.554, 10.04 and 10.12 GeV if these branching ratios ∼ 1% and less. We also demonstrate that at these photon beam energies the J/ψ energy and momentum combined distributions considered reveal distinct sensitivity to the above scenarios, respectively, at “low” J/ψ total energies and momenta, which implies that they may be an important tool to provide further evidence for the existence of the pentaquark Pci+ and Pci0 resonances and to get valuable information on their decay rates to the J/ψp and J/ψn final states. The measurements of these distributions could be performed in the future at the 12 GeV CEBAF facility.