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For k>b≥0 $k\gt b\ge 0$, let Yk,b ${Y}_{k,b}$ be the k $k$‐uniform hypergraph with two edges intersecting in b $b$ vertices. Our main result is that any n $n$‐vertex 3‐uniform hypergraph with at least n3−n−m+13+o(n3) $\left(\genfrac{}{}{0.0pt}{}{n}{3}\right)-\left(\genfrac{}{}{0.0pt}{}{n-m+1}{3}\right)+o({n}^{3})$ edges contains a collection of m $m$ vertex‐disjoint copies of Y3,2 ${Y}_{3,2}$, for m≤n∕7 $m\le n\unicode{x02215}7$. The bound on the number of edges is asymptotically best possible. This problem generalizes the Matching Conjecture of Erdős. We then use this result combined with the absorbing method to determine the asymptotically best possible minimum (k−3) $(k-3)$‐degree threshold for ℓ $\ell $‐Hamiltonicity in k $k$‐graphs, where k≥7 $k\ge 7$ is odd and ℓ=(k−1)∕2 $\ell =(k-1)\unicode{x02215}2$. Moreover, we give related results on Yk,b ${Y}_{k,b}$‐tilings and Hamilton ℓ $\ell $‐cycles with d $d$‐degree for some other values of k,ℓ,d $k,\ell ,d$. [ABSTRACT FROM AUTHOR]