The $\Gamma$-semihyperring is a generalization of the concepts of a semiring, a semihyperring and a $\Gamma$-semiring. The concepts of uniformly strongly (weak) prime $\Gamma$-semihyperring and essential extension for the $\Gamma$-semihyperring are introduced and studied some important properties in this respect. It is proved that any essential extension of a uniformly strongly prime $\Gamma$-semihyperring is a uniformly weak prime $\Gamma$-semihyperring. Also strongly prime radical of a $\Gamma$-semihyperring is introduced and its characterization is made with the help of a super sp-system. A necessary and sufficient condition for a ideal of $\Gamma$-semihyperring to be a right strongly prime ideal is provided with the help of sp-system and super sp-system.