I do not understand why larger $$p$$ will give a larger covering number.
Since when $$p\geq q$$, the corresponding hypercube is also larger (by $$\| x \| _ { q } \leq n ^ { ( 1 / q - 1 / p ) } \| x \| _ { p }$$ when $$p > q>0$$) and thus should give smaller covering set. Covering number is essentially the minimum cardinality of covering set, so it seems that larger $$p$$ should give smaller covering number?
Since you've tagged machine learning, I'll address this setting. Indeed, for the usual $$\ell_p$$ norms on $$\mathbb{R}^n$$, we have $$||x||_p\le ||x||_q$$ for $$p>q$$. However, in machine learning applications, these are normalized norms (see, e.g., Mendelson, A few notes on Statistical Learning Theory https://people.eecs.berkeley.edu/~jordan/courses/281B-spring04/readings/mendelson.ps ). Let $$D$$ be a distribution on $$\{1,\ldots,n\}$$. Then the corresponding $$\ell_p(D)$$ norm is defined by $$||x||_{\ell_p(D)}^p = \sum_{i=1}^n D(i)|x_i|^p.$$ You can verify that for these norms, the inequality goes the other way: $$||x||_{\ell_p(D)}\ge ||x||_{\ell_q(D)}$$ for $$p>q$$. This explains the covering number inequality as well.