Classical and quantum annealing is discussed in the case of a generalized kinetically constrained model, where the relaxation dynamics of a system with trivial ground state is retarded by the appearance of energy barriers in the relaxation path, following a local kinetic rule. Effectiveness of thermal and quantum fluctuations in overcoming these kinetic barriers to reach the ground state are studied. It has been shown that for certain barrier characteristics, quantum annealing might by far surpass its thermal counter part in reaching the ground state faster.