TL;DR: The bard player is arguing that lingering composition is a Performance check, and another player is stating it's Casting a Spell (Focus spell).
I have a player playing a polymath bard with lingering composition.
This player insists that when casting the composition Focus spell, on the composition’s Performance check, a critical failure is the same as a failure, and the bard doesn’t expend the Focus point (because the Focus spell requires a Performance check to determine duration, and lingering composition doesn't have a critical failure entry, only a failure entry).
My ruling as GM is that the bard is casting a Focus spell, and that the Casting a Spell rule is the active component, not the Performance check.
Lingering Composition: https://2e.aonprd.com/Spells.aspx?ID=389
Casting a Focus Spell: https://2e.aonprd.com/Rules.aspx?ID=276
You don't spend the Focus unless you succeed.
The focus spell itself define what happens at each result. If there's no "Critical failure" line then there is no effect beyond failure.
You might notice that it is a Focus spell, but unlike other ones it has a chance of not actually working, which is pretty unusual and since spending a focus point to do nothing would be terrible, then the worst case scenario is just that you miss the chance this round and must try again next round. You also "miss" the free action attached to this particular performance which prevents you to use any other metamagic focus spell (such as inspire heroics).
Seems pretty straightforward to me, specially when you consider that specific rules trumps general rules.
Lingering Composition specifically says you don't spend the focus spell if you fail. I'm not sure what the debate here is over.
When a critical failure or critical success condition is omitted they have the same result as a regular failure/success.
Which in this case means that they don't spend the focus but nothing happens either.