I agree with Alisha's answer in that yes, it could theoretically be fixed, but it's going to take some serious soldering and board rework skills to do so. Two of the solder pads have been torn off of the connector that's completely disconnected, so that wiring would have to be restored in order for it to work again. That means very carefully scraping the solder mask off the circuit board traces and soldering a piece of wire onto the bared copper then running it to where the pad used to be so the connector can be resoldered back in place.
In addition, the pads that are used for grounding and securing the connector to the board have also been torn off, so you'd need to both restore the ground signal and use some heavy duty glue to keep the connector in place since the solder connections will no longer be able to do that job.
Finally, the other connector will probably have to be replaced in order to work correctly, which will involve some fairly precise soldering skills, not to mention whether or not it's going to be easy to even find a replacement part.
Kudos to you if you feel you have the skills to tackle this repair; I might try it myself, but then again I've been working with printed circuit board for 40 years now. Otherwise, my advice would be to simply replace the circuit board. iFixit doesn't have a guide specifically for replacing that board, but the Left Joystick Replacement guide has that board being removed as part of the procedure, so if you just follow the guide and stop at Step 27, that'll be exactly what you need.
Nintendo Switch Lite Left Joystick Replacement - iFixit Repair Guide
The board itself isn't all that expensive; I found an example on Amazon for only about $10 USD; there are many other places you can find them such as eBay and AliExpress among others.
Amazon.com: Left Side Motherboard Nintendo Switch Lite Replacement PCB L Button Professional Game Console Key Board Part for Handle : Video Games
Personally, for the money that's the way I'd go rather than taking a chance on a repair that only has a moderate chance of success.
Good luck; let us know how it turns out!
P.S. I neglected to mention this, but the part you need to replace isn't actually considered the motherboard, rather most sites are calling it a daughter board, as it's separate from the motherboard and, more importantly, much much cheaper.