Manin's conjecture over finite fields predicts the asymptotic formula for the counting function of rational curves of bounded degree on smooth Fano varieties defined over finite fields. In his unpublished notes, Batyrev developed a heuristic for this conjecture and the assumptions he used are generalized and systemized as Geometric Manin's conjecture in characteristic 0. In this talk I would like to explain our attempt to understand Geometric Manin's conjecture in characteristic p for weak del Pezzo surfaces extending results on GMC for del Pezzo surfaces in char 0 by Testa to char p for most primes p. In the course of our investigation, we observe that some pathological examples of weak del Pezzo surfaces studied by birational geometers provide us examples of weak del Pezzo surfaces whose exceptional sets for weak Manin's conjecture are Zariski dense which is contrast to some positive results on exceptional sets in char 0. This is joint work with Roya Beheshti, Brian Lehmann, and Eric Riedl.