Finite groups with formational subnormal primary subgroups of bounded exponent

Let $\mathfrak{U}_k$ be the class of all supersoluble groups in which exponents are not divided by the $(k+1)$-th powers of primes. We investigate the classes $\mathrm{w}\mathfrak{U}_k$ and $\mathrm{v}\mathfrak{U}_k$ that contain all finite groups in which every Sylow and, respectively, every cyclic primary subgroup is $\mathfrak{U}_k$-subnormal. We prove that $\mathrm{w}\mathfrak{U}_k$ and $\mathrm{v}\mathfrak{U}_k$ are subgroup-closed saturated formations and obtain the characterizations of these formations.