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 λ₁x₁²+⋯ +λₙxₙ²≡ λₙ₊₁pᵐ, where p is a fixed odd prime, λ₁,...,λₙ₊₁ are integer coefficients such that (λ₁⋯ λ ₙ,p)=1 and m→ ∞. If n≥ 6, p≥ 5 and the coefficients are fixed and satisfy λ₁,...,λₙ>0 and (λₙ₊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+ε)m, centered at the origin. If n≥ 4 and λₙ₊₁=0 (homogeneous case), we prove a result of the same strength for coefficients λᵢ which are allowed to vary with m. These results extend previous results of the first- and the third-named authors and N. Bag.