In this paper we show that for $\ell < k$ satisfying $(k-\ell)\nmid k$, $(p,\mu)$-denseness plus a minimum $(\ell+1)$-vertex-degree $\alpha n^{k-\ell-1}$ guarantees Hamilton $\ell$-cycles, but requiring a minimum $\ell$-vertex-degree $\Omega(n^{k-\ell})$ instead is not sufficient. This answers a question of Lenz--Mubayi--Mycroft and characterizes the triples $(k,\ell, d)$ such that degenerate choices of $p$ and $\alpha$ force $\ell$-Hamiltonicity. We actually prove a general result on $\ell$-Hamiltonicity in quasi-random $k$-graphs, assuming a minimum vertex degree and essentially that every two $\ell$-sets can be connected by a constant length $\ell$-path. This result reduces the $\ell$-Hamiltonicity problem to the study of the connection property.