Here's one for posterity.

I have a portable DVD player (10" Mintek MDP-1060) has gotten into a
situation where it will spin up but then not detect disc. I've heard
suggestions of shot laser, clean the lens, laser re-alignment, and I'm
sure there are situations where that is valid.

Part of my problem is an aggressive 3-year old coupled w/ a cheap DVD
player, but that's a separate matter.

On 1 prior occasion was able to get the thing to work by trying an
audio cd instead.

On all occasions the issue was (I think) the play head / pickup was way
out at the edge of its read area.

After trying all sorts of things (nudging the head transport gingerly
included), I simply put a disc in and put the thing on its side
opposite the head (in my case the head was on the left of the unit so
put the unit on it's right side - ie, resting on its right edge,
perpendicular to normal play setting. I did this thinking that gravity
could give the thing or the laser the nudge it needed to get back to
norm.

Then, I pressed play, and bingo, it worked.

Good luck in the future if you read this and hopefully it'll solve your
problem in lieu of trashing the unit.