For n >= 1 and p >= 3 define an (n,p)-polyhedron as a convex polyhedron with exactly n congruent regular p-gonal faces. No restrictions on the other faces.
For n = 1, you can use pyramids with regular p-gonal bases. For n = 2, uniform prisms or uniform antiprisms with regular p-gonal bases.
For p = 3, you can find an example for n = 1, 2, ..., 20 by doing the appropriate truncation at n vertices of a regular dodecahedron. (There are of course other examples.) Antiprisms can be used for all even n > 20. J24 & J26 take care of n = 25, and J47 & J71 take care of n = 35.
For p = 4, use the uniform n-prism for n >= 3.
For p = 5, use the appropriate truncations at n vertices of the regular icosahedron for n = 1, 2, ..., 12.
For p = 6, the truncated icosahedron has n = 20 regular hexagons. Augment with hexagonal pyramids with heights so that the resulting polyhedron remains convex. You can do this to get (n,6)-polyhedra for for n = 1, 2, ..., 19.
For p = 8, the truncated cube has n = 6 regular octagons. Augment with octagonal pyramids with heights so that the resulting polyhedron remains convex. You can do this to get (n,8)-polyhedra for n = 1, 2, 3, 4, 5.
For n = 10, the truncated dodecahedron has n =12 regular decagons. Augment with decagonal pyramids with heights so that the resulting polyhedron remains convex. You can do this to get (n,10)-polyhedra for n = 1, 2, ..., 11.
Question 1: Are there examples for p =3 and odd n > 20 besides n = 25 and n = 35?
Question 2: Are there examples for p = 5 and n > 12?
Question 3: Are there examples for p = 6 and n > 20?
Question 4: Are there examples for p = 8 and n > 6?
Question 5: Are there examples for p = 10 and n > 12?
Question 6: Are there examples for p = 7, 9, 11, 12, 13, ... and n> = 3?