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 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 construction of a finitely presented non-amenable torsion-by-cyclic group is based on 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).