  This paper introduces group-cograded monoidal Hom-Hopf algebras, and shows
that this kind of group-cograded monoidal Hom-Hopf algebras are monoidal
Hom-Hopf algebras in the Turaev category $\mathcal{J}_{k}$ introduced by
Canepeel and De Lombaerde. Then we define the $p$-Yetter-Drinfeld category over
a group-cograded monoidal Hom-Hopf algebra, and construct a new kind of braided
$T$-categories.
