Can you give me an example of a finitely generated infinitely presented amenable group which is a quotient of a finitely presented amenable group?
Take the finitely presented solvable group $G$ with undecidable word problem, constructed by Kharlampovich. That group has infinite center that is a direct product of infinite number of cyclic group. The center has uncountably many subgroups $N_\alpha$, each normal in $G$, most groups $G/N_\alpha$ are not finitely presented but amenable. The description of an easier construction of Kharlampovich's group is in our survey Kharlampovich, O. G. Sapir, M. V. Algorithmic problems in varieties. Internat. J. Algebra Comput. 5 (1995), no. 4-5, 379–602. Another, easier, example, is Abels' group and its quotients by central subgroups. Abels' group can be found here: H. Abels, An example of a finitely presented solvable group, Homological group theory (Proc. Sympos., Durham, 1977), London Math. Soc. Lecture Note Ser., 1979, p. 205–211. It has been mentioned on MO before (see Cornulier's answers).