Let us call an "HNN extension" with non-injective homomorphism, between associated subgroups ni-HNN extension as opposite to the ordinary HNN extensions. I know three examples where ni-HNN extensions were used.
Ilya Kapovich result that an ascending ni-HNN extension of a free group $F$ with non-injective homomorphism $\phi : F\to F$ is actually isomorphic to an ascending HNN extension of some free group ("Mapping tori of endomorphisms of free groups", Comm. Algebra 28 (2000), no. 6, 2895–2917).
A result of Igor Lysenok that the Lysenok finitely presented group containing the Grigorchuk group is a ni-HNN extension of some finitely presented group ("A set of defining relations for the Grigorchuk group", Mat. Zametki 38 (1985), no. 4, 503–516).
Our with A. Yu. Olshanskii finitely presented non-amenable torsion-by-cyclic group is a ni-HNN extension of some finitely presented group containing the free Burnside group ("Non-amenable finitely presented torsion-by-cyclic groups." Publ. Math. Inst. Hautes Études Sci. No. 96 (2002), 43–169).