In the present paper, $t$-generalized supplemented modules are defined starting from the generalized ⨁-supplemented modules. In addition, we present examples separating the $t$-generalized supplemented modules, supplemented modules, and generalized ⨁-supplemented modules and also show the equality of these modules for projective and finitely generated modules. Moreover, we define cofinitely $t$-generalized supplemented modules and give the characterization of these modules. Furthermore, for any ring $R$, we show that any finite direct sum of $t$-generalized supplemented $R$-modules is $t$-generalized supplemented and that any direct sum of cofinitely $t$-generalized supplemented $R$-modules is a cofinitely $t$-generalized supplemented module.
