Can I legally DJ with purchased iTunes music in public?

From the terms of use:

"You may use the Services and Content only for personal, noncommercial purposes"

so per the license, no, you cannot use tracks from the iTunes Store in an commercial setting. You would need to purchase any tracks you want to play either on CD or from a site, if any such exist, that doesn't have that restriction. You also will need a license, as Zurarczurx said, to play any such tracks publicly.

Regards.

In the UK - probably not, possibly a grey area.

The copyright for the music still rests with the original artist and (in the UK) as long as you or the venue has a PPL PRS licence then you can play copyrighted music.

The problem is that I think there's a clause in the iTunes agreement that says you can only use the music for personal use. You should check the agreement you clicked on in iTunes.

A google search finds.

PPL and PRS for Music are two separate collective management organisations (CMOs). PPL collects and distributes money on behalf of performers and record companies for the use of their recorded music. PRS for Music collects and distributes money on behalf of songwriters, composers and music publishers, for the use of their musical compositions and lyrics.

In 2018 PPL and PRS for Music joined forces to streamline part of what we do, coming together to launch

PPL PRS Ltd.

This joint venture means that businesses and organisations playing or performing music in public can now obtain a single licence, TheMusicLicence.

PPL and PRS for Music continue to set their respective public performance tariffs separately.

"Google" is a popular website that allows people to search for information.

I can't comment on what your leisure centre or gym does nor why it does it.

I’m still confused . Most leisure centres has a ppl licence -or two one for studios or gym and another for the bar etc

but some gyms ask the instructor to pay £1.32 a class for prs do you need both ppl and prs

In the US, organizations and businesses can purchase a blanket license valid for 1 year. I suspect your gym has a similar license.