  Given two $r$-uniform hypergraphs $F$ and $H$, we say that $H$ has an
$F$-covering if every vertex in $H$ is contained in a copy of $F$. Let
$c_{i}(n,F)$ be the least integer such that every $n$-vertex $r$-graph $H$ with
$\delta_{i}(H)>c_i(n,F)$ has an $F$-covering. Falgas-Ravry, Markst\"om and Zhao
(Combin. Probab. Comput., 2021) asymptotically determined
$c_1(n,K_{4}^{(3)-})$, where $K_{4}^{(3)-}$ is obtained by deleting an edge
from the complete $3$-graph on $4$ vertices. Later, Tang, Ma and Hou (arXiv,
2022) asymptotically determined $c_1(n,C_{6}^{(3)})$, where $C_{6}^{(3)}$ is
the linear triangle, i.e. $C_{6}^{(3)}=([6],\{123,345,561\})$. In this paper,
we determine $c_1(n,F_5)$ asymptotically, where $F_5$ is the generalized
triangle, i.e. $F_5=([5],\{123,124,345\})$. We also determine the exact values
of $c_1(n,F)$, where $F$ is any connected $3$-graphs with $3$ edges and
$F\notin\{K_4^{(3)-}, C_{6}^{(3)}, F_5\}$.
