The Breit equation with two electromagnetic form factors is studied to obtain a potential with finite-size corrections. This potential with proton structure effects includes, apart from the standard Coulomb term, the Darwin term, retarded potentials, spin–spin and spin–orbit interactions corresponding to the fine and hyperfine structures in the hydrogen atom. Analytical expressions for the hyperfine potential with form factors and the subsequent energy levels including the proton structure corrections are given using the dipole form of the form factors. Numerical results are presented for the finite-size corrections in the 1S and 2S hyperfine splittings in the hydrogen atom, the Sternheim observable D21 and the 2S and 2P hyperfine splittings in muonic hydrogen. Finally, a comparison with some other existing methods in the literature is presented.