For an $r$-uniform hypergraph $H$, let $ν^{(m)}(H)$ denote the maximum size of a set~$M$ of edges in $H$ such that every two edges in $M$ intersect in less than $m$ vertices, and let $τ^{(m)}(H)$ denote the minimum size of a collection $C$ of $m$-sets of vertices such that every edge in $H$ contains an element of $C$. The fractional analogues of these parameters are denoted by $ν^{*(m)}(H)$ and $τ^{*(m)}(H)$, respectively. Generalizing a famous conjecture of Tuza on covering triangles in a graph, Aharoni and Zerbib conjectured that for every $r$-uniform hypergraph $H$, $τ^{(r-1)}(H)/ν^{(r-1)}(H) \leq \lceil{\frac{r+1}{2}}\rceil$. In this paper we prove bounds on the ratio between the parameters $τ^{(m)}$ and $ν^{(m)}$, and their fractional analogues. Our main result is that, for every $r$-uniform hypergraph~$H$, \[ τ^{*(r-1)}(H)/ν^{(r-1)}(H) \le \begin{cases} \frac{3}{4}r - \frac{r}{4(r+1)} &\text{for }r\text{ even,}\\ \frac{3}{4}r - \frac{r}{4(r+2)} &\text{for }r\text{ odd.} \\ \end{cases} \] This improves the known bound of $r-1$. We also prove that, for every $r$-uniform hypergraph $H$, $τ^{(m)}(H)/ν^{*(m)}(H) \le \operatorname{ex}_m(r, m+1)$, where the Turán number $\operatorname{ex}_r(n, k)$ is the maximum number of edges in an $r$-uniform hypergraph on $n$ vertices that does not contain a copy of the complete $r$-uniform hypergraph on $k$ vertices. Finally, we prove further bounds in the special cases $(r,m)=(4,2)$ and $(r,m)=(4,3)$.