Suppose that $N$ is a sufficiently large real number. In this paper it is proved that for $2<c<\frac{990}{479}$, the Diophantine equation

$$\left[p_1^c\right]+\left[p_2^c\right]+\left[p_3^c\right]+\left[p_4^c\right]+\left[p_5^c\right]=N$$

is solvable in primes $p_1,p_2,p_3,p_4,p_5$ such that each of the numbers $p_i+2,i=1,2,3,4,5$ has at most $\left[\frac{6227}{3960-1916c}\right]$ prime factors.