Working in a music store and getting to plug everything in, *most* of the time the designated speakers are very hard to replicate in terms of quality (The Korg PA1000 has the best speakers I've ever heard for a KB, and using most portable PA's under $1500 won't even come close). The speakers of the PSRsx900 are even clearer (but don't have the built in sub) and the PSRs970/975 have fantastic speakers, just a smidge less warm than the 2 aforementioned heavyweights.
So if you plug the s970 into a pair of... well, without naming specific brands and models, but let's just say if I plug it a pair of almost *any* live sound reinforcement speakers under $800/pair, it will sound much, much worse than the on board. At around the $1100/pair price point, there are 2 that would sound as good as the built in speakers (slightly warmer, but also with less bass).
I can say I miss the mark 1 versions of the QSC K series... they had a built in 'deep bass' mode that really, really worked (*without* frapping the speaker out... most speakers when you try to add bass, they basically introduce a farting noise)
Now, I *can* say that a current QSC set of K10.2 or K12.2 would sound amazing, as long as you added a sub. Or JBL Eons (but anyone who can hear above 15kz will find them not as clear and open as the onboard speakers).
So yes, if it sounds worse through a PA, that's actually very common for a lot of models of PA speakers. Is there a fix without purchasing better speakers? Unfortunately, not really... you can EQ it all you want, but a lot of speaker frequency response curves aren't designed to be remotely flat (which is what a full frequency range instrument like a keyboard or electronic drum kit demands), meaning if it sounds muddy because the speakers can't flatly reproduce any frequencies above say, 12 kHz, no amount of compensating will bring those frequencies back.
I suppose you could just purchase a separate PA system for yourself. Out of curiosity, what are you running through? Also keep in mind, your PSR is designed for a stereo image, but most people playing guitars, bass guitar, and singing through a PA usually design the sound field in mono. Concerts are usually mono sound sourced into a multiple speaker array, so they can 'predict' accurately what each audience member hears. We use an 20 PA speaker array that's fed from 1 mono channel. It doesn't sound mono because they're pointing in all directions.
The point is, if they're designing their sound field in mono, stereo sampled instruments aren't optimised acoustically for that (that's why most pro stage pianos include a mono piano sound, specifically for live gigging). THEN you can send those to the master mixer with everyone's else's signals, and it won't sound so thin (stereo designed signals, even if you use the L+R into a single channel, always sound thinner than mono designed signals when doing a live total mix, hence why pro stage pianos have mono piano options).
Luckily you *can* download samples designed for mono use into the s970.
Mark