Small solutions to inhomogeneous and homogeneous quadratic congruences modulo prime powers

We prove asymptotic formulae for small weighted solutions of quadratic congruences of the form ₁x₁²+ +ₙxₙ² ₍+₁pᵐ, where p is a fixed odd prime, ₁,. . . , ₍+₁ are integer coefficients such that (₁ ₍, p) =1 and m. If n 6, p 5 and the coefficients are fixed and satisfy ₁,. . . , ₙ>0 and (₍+₁, p) =1 (inhomogeneous case), we obtain an asymptotic formula which is valid for integral solutions (x₁,. . . , xₙ) in cubes of side length at least p^ (1/2+) m, centered at the origin. If n 4 and ₍+₁=0 (homogeneous case), we prove a result of the same strength for coefficients ᵢ which are allowed to vary with m. These results extend previous results of the first- and the third-named authors and N. Bag.