A kinetic model is developed to describe the growth of crystals under the influence of foreign particles in terms of heterogeneous two-dimensional nucleation. In the context of this model, the free energy barrier of two-dimensional nucleation in the presence of foreign particles and the kinetics for the nucleation and growth are examined theoretically. It follows that the contact angle, size and density of adsorbed foreign particles play a crucial role in controling the 2D nucleation barrier and growth kinetics. Based on our model, many crucial experimental findings, such as dust-induced surface roughening and the various kinetics of dislocation-free growth, are properly interpreted. The promotion effect of foreign particles on crystal growth is also analyzed from the view of designing additives. To our knowledge, this is the first systematic consideration of effects of foreign particles on the 2D nucleation process and growth kinetics, and the model generally covers both heterogeneous and conventional homogeneous 2D nucleation growth.