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A PN formalism (for post-Newtonian formalism) is a method of representing dynamics as Newton's laws along with a term (or terms) covering the differences between Newtonian mechanics and the effects of general relativity (GR). A perfect fit would be nice but generally an approximation of GR is what is attainable which may be sufficiently accurate to be useful (and the term post-Newtonian approximation is also used). Variants of this notion have been used since GR's invention because GR calculations can be extremely difficult to carry out in a straightforward manner so good approximations have value: Einstein used such methods from the beginning. The PN formalism uses an expansion such that approximations can be made using one, two, or more terms, such versions labeled 1PN, 2PN, 3PN, etc.
A post-Newtonian correction term (PN correction term) is a term added to a (Newtonian) formula/equation to include or approximate the effects of relativity.