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. 

1. Ilya Kapovich [result][1] 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. 

2. A result of Igor Lysenok and Rostislav Grigorchuk that a finitely presented elementary amenable 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, "An example of a finitely presented amenable group that does not belong to the class EG". Mat. Sb. 189 (1998), no. 1, 79--100). 

3. 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). 


  [1]: https://arxiv.org/pdf/math/0208189.pdf