  We employ the absorbing-path method in order to prove two results regarding
the emergence of tight Hamilton cycles in the so called {\em two-path} or {\em
cherry}-quasirandom $3$-graphs.
  Our first result asserts that for any fixed real $\alpha >0$,
cherry-quasirandom $3$-graphs of sufficiently large order $n$ having minimum
$2$-degree at least $\alpha (n-2)$ have a tight Hamilton cycle.
  Our second result concerns the minimum $1$-degree sufficient for such
$3$-graphs to have a tight Hamilton cycle. Roughly speaking, we prove that for
every $d,\alpha >0$ satisfying $d + \alpha >1$, any sufficiently large
$n$-vertex such $3$-graph $H$ of density $d$ and minimum $1$-degree at least
$\alpha \binom{n-1}{2}$, has a tight Hamilton cycle.
