In this paper, n-fold branched coverings of a closed nonorientable surface ${\mathcal S}$ of genus p with $r\geq 1$ cyclic branch points (that is, such that all ramification points over them are of multiplicity n) are considered. The number N p,r (n) of such coverings up to equivalence is evaluated explicitly in a closed form (without using any complicated functions such as irreducible characters of the symmetric groups). The obtained formulas depend on the parity of r and n. The method is based on some previous enumerative results and techniques for nonorientable surfaces. In particular, we generalize the approach developed for the counting of unbranched coverings of nonorientable surfaces and make use of the analytical method of roots-of-unity sums.