Sparse H-colourable graphs of bounded maximum degree

Let F be a graph of order at most k. We prove that for any integer g there is a graph G of girth at least g and of maximum degree at most 5k(13) such that G admits a surjective homomorphism c to F, and moreover, for any F-pointed graph H with at most k vertices, and for any homomorphism h from G to H there is a unique homomorphism f from F to H such that h = f circle c. As a consequence, we prove that if His a projective graph of order k, then for any finite family F of prescribed mappings from a set X to V(H) (with \F\ = t), there is a graph G of arbitrary large girth and of maximum degree at most 5k(26mt) (where m = \X\) such that X subset of or equal to V(G) and up to an automorphism of H, there are exactly t homomorphisms from G to H, each of which is an extension of an f is an element of F.