Icyo
I've never critiqued a Yoshi match before so I might have some bad advice, but I'll do my best!
GAME ONE:
1) You need to hesitate less. The Puff was able to smack you around partly because you crouched so frequently and were too spooked to attack when you did get an opening. Keep in mind that bair, while a great spacing tool in general, is unreliable agaist Puff due to her superior reach. In general, I find the best way to get to a Puff is to get her vertically: while her bair has incredible reach, her dair is significantly more risky (and Yoshi's u-air is wonderful).
2) At 0:25 or so you're knocked off stage and attempt to throw an egg, I believe to grab that platform? Whenever off stage with Yoshi against anyone, your very first instinct should be recovery. Don't lose that double jump! There's very little excuse for dying at 50%, and no reason to go for ledge when your opponent is that close to you. At around 0:50, you go for a down-b to ledge and miss it. The thing is you'd already almost made it back to stage! In this situation a wave land onto the left platform would have meant getting your jump back and maybe even (if you slid off the platform) getting in a dair on an unsuspecting Puff). This is again about confidence in your character and yourself.
3) At 0:43 You throw an egg in neutral. You do this quite a bit and almost every time it's done when Puff is extremely close to you. Eggs do a lot of damage and are great against Puff but knowing when not to use them is super important, and you missed a couple kill opportunities because of egg throw. Trust me when I say I know your pain here because I also throw hella eggs in neutral and I too suffer for it.
4) This is sort of a continuation on part 1 here, but those crouches are really hurting you. As the match goes on, the Puff gets a better idea of how to deal with it, going from hitting you for decent damage with bairs to the inevitable rest combo at around 1:17. This is maybe a worse habit than the egg toss; I don't believe it works out in your favor even once in game one. Fortunately, it's a very easy habit to fix. I personally suggest replacing your crouches with your dash dance whenever possible. Your dash dance is better than mine, I'd argue.
5) This one is minor, but at 1:56 you go for a downsmash, which knocks Puff too far away for you to follow up. I do think this part is all about nerves.
GAME 2:
1) In Game 2 you become a whole different person or Owl. You're dash dancing, you're not afraid to approach. It shows, you immediately start scaring the Puff and you gain a percent lead for it. Yet, every time you're about to get the big money, you go for another egg throw, which almost always gets you punished. In general, you may need to start thinking more about your match in between your stocks. Note that your upsmashes were destroying this Puff. If you hadn't stood there crouching, or attempted eggs during your combos, I'm fairly certain you would have won.