What can I put underneath my mattress to make it softer?

If you can feel the plywood, replace the mattress.  Full stop.  The plywood is not and never was the problem.

A box spring mattress would do the trick. You could also try and find a large piece of foam, or just get a second mattress. 

Yes--another mattress. Yours simply isn't thick enough or soft enough for you. You can also use a slab of foam rubber, if you can find one--they are hard to find in thicknesses over 2". (And expensive.  A cheap, additional mattress would be much less expensive.) 

Your mattress seems like a foam instead of innerspring. The problem is probably the mattress. Foam mattresses are cheap, but provide no support. Nothing will fix it but a different mattress if it's a foam one.

Sounds like your mattress is a foam one. It isn't thick enough, or isn't dense enough, or maybe both. If you can feel the plywood through it, it's not adequate for someone your size. Foam that isn't dense enough squashes down too much, and that also happens if it isn't thick enough. If a coil mattress is squashing down like that it must be a really really cheap one. If you put some really dense foam at least 3" thick on top or underneath, that might do it. But it might not. Your best bet is really to buy another mattress that is right for you. Go to a place where you can try them out before buying.

Any padding will help, although if the mattress is that sh*tty, ypu might as well replace it

Try using a padded mattress cover on top not underneath

Was the mattress intended to be assembled with a box spring?If so I then that is what you probably have to do - buy a box spring .

Otherwise perhaps removing the plywood base and replacing it with wooden SLATS would be beneficial to soften The feel of the mattress . Also, there are thick mattress pads that may soften your bed without the need of an actual topper.

A box spring.

But honestly, if you can feel that through the mattress you need a different mattress.
Mattresses are expensive but properly selected and cared for will last at least a couple of decades.  I have a wonderful mattress and box spring I paid a couple thousand dollars for 20 years ago and I still love it.

You an also find excellent deals on craigslist, although be very very careful with used mattresses because they can have gross stuff on them, including bedbugs.  I am setting up my extra room as a bedroom and found a $2,400 mattress on craigslist for free.  It was used for about a year by a single lesbian (no sex on the mattress) then sealed up in storage for 7 years while she travelled and worked internationally.  Now she had a partner and bigger bed, so this ended up on craigslist.  It is the teeniest bit musty from storage, has been sitting in my covered patio for a few days airing out and the must is almost gone.  I haven't even done the sunlight/baking soda/vaccumm thing yet.  My friend moving into that room is totally stoked. 

nothing but you could put a matress topper on top that will make it softer 

try stuffing your mattress with human hair! light as a cloud!

ADD another topper to ease the pain.

If you stuffed the mattress with money, you would love the lumps.

Foam like 3" of firm foam they use for camping.  I got that on top of my box spring mattress to make it softer.  That works fine. Or make a hammock?  Wood has never been soft.  Softwood implies it is from a needle tree, you can stick your fingernail into the board and leave a mark. Hardwood is a tree that loses its leaves every winter such as maple, oak, teak and is harder than the fingernail, so used for tables, cutting boards, furniture...as it looks prettier than soft wood, as it has intricate grain patterns to the board.
.  The wood you want to lay on is a pile of sawdust.  That is soft.

Not under.  Over.  Go to any store that sells mattresses and buy the foam topper in your bed size.  It is about three inches think.  Also go to any second hand and buy a used bed spring box to set on those plywood boards. 

It's called a bedspring. 

The mattress is rubbish. Time to invest in a good one.