We prove asymptotic formulae for small weighted solutions of quadratic congruences of the form \(\lambda _1x_1^2+\cdots +\lambda _nx_n^2\equiv \lambda _{n+1}\bmod {p^m}\) , where p is a fixed odd prime, \(\lambda _1,...,\lambda _{n+1}\) are integer coefficients such that \((\lambda _1\cdots \lambda _{n},p)=1\) and \(m\rightarrow \infty \) . If \(n\ge 6\) , \(p\ge 5\) and the coefficients are fixed and satisfy \(\lambda _1,...,\lambda _n>0\) and \((\lambda _{n+1},p)=1\) (inhomogeneous case), we obtain an asymptotic formula which is valid for integral solutions \((x_1,...,x_n)\) in cubes of side length at least \(p^{(1/2+\varepsilon )m}\) , centered at the origin. If \(n\ge 4\) and \(\lambda _{n+1}=0\) (homogeneous case), we prove a result of the same strength for coefficients \(\lambda _i\) which are allowed to vary with m. These results extend previous results of the first- and the third-named authors and N. Bag.
