At subzero temperatures, cloud particles can contain both ice and liquid water fractions. Wet growth of precipitation particles occurs when supercooled cloud liquid is accreted faster than it can freeze on impact. With a flexible framework, the theory of wet growth of hail is extended to the case of the inhomogeneities of surface temperature and of liquid coverage over the surface of the particle. The theory treats the heat fluxes between its wet and dry parts and radial heat fluxes from the sponge layer through the liquid skin to the air. The theory parameterizes effects of nonsphericity of hail particles on their growth by accretion. Gradual internal freezing of any liquid soaking the hail or graupel particle's interior during dry growth ("riming") is treated as well. In this way, the microphysical recycling envisaged by Pflaum in a paper in 1980 is treated, with alternating episodes of wet and dry growth. The present paper, the first of a two-part paper, describes the scheme to treat wet growth, accounting for dependencies on condensate content, temperature, and particle size. Comparison with the laboratory experiments is presented.