If they can work out a partnership, sure why not.

Especially if they could work with Nintendo and score some of the component deals on the next handheld (LCD screen, cameras, battery). Thats good for Sony, good for Nintendo. 

It's not really "third party" in a traditional sense though ... there is no alternative here, it's Nintendo or make silly/cheap smartphone games.

Sony did make some good money off the PSP and some of their franchises did sell well on the PSP ... Gran Turismo sold 4.6 million, God of Wars 3.5+ million ... that's not charity money.

Sony's IP, especially the Western ones, would fill out Nintendo's handheld portfolio nicely too since Nintendo lacks in Western IPs on the handheld side badly. Things like God Of War, Uncharted, MLB, Last Of Us, Gran Turismo, Killzone definitely would bring a different (and welcome IMO) dynamic to Nintendo. 

I think if Sony threw more support behind Nintendo other third parties like EA and Square-Enix and Konami would also follow suit. Perhaps Street Fighter V could also be allowed on the HDS (after the PS4 version of course). Works for me. 

They'd still obviously make games for the PS4/PS5. They could just keep their foot in the handheld market too. PSP was at one time a nice part of their business, I don't think it really helps Sony to lose that entirely. 

Sony has no problem funding studios to make projects, so IP like Uncharted and GOW can be outsourced, that's no the problem. The problem for Sony was they didn't have a userbase with Vita to sell those games to, if they did they would've continued to support the Vita. But by supporting Nintendo they're assured a decent userbase to sell to, like the PSP had. 

If they expand the deal to get Nintendo to bring some of their IP to the PS4 .... bang. That's a huge win for them. 

That could actually be a really good idea! It'd be a great way for both of them to organise a mutually sweet deal. 

But do Sony trend towards doing deals that trade favours accross their different sectors?

This, with psnow you will see sony on everything sooner or later they sure wont waste time and money devoloping for a handheld if they can just stream

There isn't really any downside for Sony to do this.

If they can score a supplier deal with Nintendo in the process (ie: to supply the LCD or other components), they would do it on that basis alone IMO.

But there are other benefits from Sony's POV. Nintendo handhelds attract the youth audience, so having their games on a Nintendo handheld can introduce kids to many of Sony's IP before they can maybe afford that $400 PS4. 

They're not competing with Nintendo anymore, so why not.

Would have about as much impact as third party support has on the WiiU.

I think it would have more impact than people think. 

If Sony supports, than I think a number of other third parties would throw heavier support on the Nintendo handheld and it would be seen more of a industry standard rather than a "kids toy" that Western developers look at the DS/3DS as. 

You probably would start seeing more Western developer content for the device than the 3DS or DS got or other traditionally Sony-centric IP like Final Fantasy, Metal Gear, Borderlands, Grand Theft Auto, FIFA, and Street Fighter may also start to become more prevalent. 

Part of Nintendo's problem is they don't themselves really try to engage in these other types of genres, so third parties look at that and say "well if Nintendo's not trying why should we? They're making a bunch of mascot games for it so we'll support it with mascot and kids games". 

With Sony onboard though they could act as buffer a developer like Rockstar could say "well hey Sony is making an Uncharted and God of War for this thing, maybe we can try a Grand Theft Auto for it". 

I think this is the crux of my point. Also I don't believe Sony would have any influence on third party developer's predilections in the handheld space.

Grand Theft Auto, Final Fantasy Dissidia, FIFA, God of War, Gran Turismo, etc. and others have sold well on the PSP in the past. 

The issue is today is Nintendo is the only one strong enough to push through smartphones and have a handheld userbase that makes selling big IP titles feasible. 

Third parties trust Sony, if Sony banks on Nintendo, then a lot of them would follow suit too IMO. Sony brings legitimacy and wider market demographic to the table. 

Sony main 1st party studios have too much pride (with sprinkles of arrogance) to develop handheld games.