If one tried to write a term in a Lagrangian which only used one field (for example, in QED if one tried to write a term which looked like $\gamma^{\mu} A_{\mu}$), would this term have any meaning in terms of Feynman rules?
I leave aside whether the term would be forbidden because of violation of gauge invariance and other principles, and ask only if it would make sense in theory to have such a term when writing down Feynman diagrams?
It seems that a term in the Lagrangian that only involves one field is not very meaningful in terms of Feynman rules.
Feynman Rules in QED
Rule 0: a factor of $i$.
Rule 1: External lines: We attach wave functions or polarizations to each incoming or outgoing particle.
Rule 2: Internal lines: To each internal line, we attach a propagator depending on particle species.
Rule 3: Vertices: To each vertex, we attach a coupling constant and a factor depending on the type of interaction.
Rule 4: Loops: To each closed loop, we attach a factor of $(2π)^4$ and an integral over the loop momentum.
As you can see, the Feynman rules involve terms that have at least two fields (either external or internal) and at least one vertex. A term that only has one field would not contribute to any Feynman diagram, and therefore would not have any physical significance. In fact, such a term would violate gauge invariance and other principles, as you mentioned. For example, in QED, a term like $γ^μA_μ$ would not be invariant under the gauge transformation $A_μ→A_μ+∂_μλ$, where $λ$ is an arbitrary function. Therefore, such a term is not allowed in the Lagrangian.
