  For a positive integer $m$ and a finite non-negative Borel measure $\mu$ on
the unit circle, we study the Hadamard multipliers of higher order weighted
Dirichlet-type spaces $\mathcal H_{\mu, m}$. We show that if
$\alpha>\frac{1}{2},$ then for any $f$ in $\mathcal H_{\mu, m},$ the sequence
of generalized Ces{\`a}ro sums $\{\sigma_n^{\alpha}[f]\}$ converges to $f$. We
further show that if $\alpha=\frac{1}{2}$ then for the Dirac delta measure
supported at any point on the unit circle, the previous statement breaks down
for every positive integer $m$.
