Let \(\mu \) be a finite positive Borel measure on [0, 1) and let \(H(\mathbb {D})\) be the space of all analytic function in the unit disc \(\mathbb {D}\). The Cesàro-like operator \(\mathcal {C}_\mu \) is defined in \(H(\mathbb {D})\) as follows: If \(f \in H(\mathbb {D})\), \(f(z)=\sum _{n=0}^{\infty }a_{n}z^{n} (z\in \mathbb {D})\), then

where for \(n\ge 0\), \(\mu _n\) denotes the n-th moment of the measure \(\mu \), that is, \(\mu _n=\int _{0}^{1} t^{n}d\mu (t)\). For \(s > 1\), let X be a Banach subspace of \(H(\mathbb {D})\) lying between the mean Lipschtz space \(\Lambda ^{s}_{\frac{1}{s}}\) and the Bloch space \(\mathcal {B}\). In this paper we characterize the measures \(\mu \) as above for which \(\mathcal {C}_\mu \) is bounded (compact) from X into any of the Hardy spaces \(H^{p} ~ (1\le p\le \infty )\).