A strange call on a punt- HELP!

[Welpe]

Yes he can. I assume you're talking Federation rules.

Rule reference:

4-2-2

The ball becomes dead and the down is ended:

f. When the kickers catch or recover any free kick anywhere, and when the
kickers catch or recover a scrimmage kick beyond the neutral zone and
when the kickers are first (i.e., before any touching by the receivers) to
touch a scrimmage kick after it has come to rest beyond the neutral zone
and between the goal lines.

There is no prohibition against Team K advancing a scrimmage kick recovered behind the neutral zone.