How can I build a cheap playhouse for a 4-year old?

This isn't exactly "building", but I'm seeing tons of super- cute older plastic playhouses that people have painted. Check out this post: https://thestir.cafemom.com/toddlers_preschoolers/213104/costco-playhouse-hacks/238496/before_amp_after_perfection/7

Check your local Facebook Marketplace or yard sales. This is the time of year when people are cleaning out their garages. Good luck!

I made one for my granddaughter out of two cardboard boxes I got from a friend who had a frame shop. . . glass came in these boxes so they were sturdy. It was lot of fun, but of course not too long lasting (I gave it away to a child at one of my yard sales)

You can often get large boxes from a furniture store. And I have found the little ones love decorating it. When I cut out a window I left it “hinged” on both sides and hot glued handles made of cardboard to open and close them. There are endless ways to decorate and make it one of a kind. Kids love them and when they outgrow it ....easy to get rid of with no real cost.