As a Kat main on season 4.... I think she needs mana

1- She is all point and click.

This may see not very clear to why she should have mana, but is pretty simple : she will not use abilities on you except if it is on CD. But since she can't miss ( maybe her W , but whatever ), she will always damage you except if she needs to push the wave.

2- This would stop her from resetting a lot early game

Katarina only snowballs hard because she can reset a lot early game, especially because she has no mana problems. Not only that, but ganking bot lane and getting 2 assists puts your ult CD in a very low CD, which opens low opportunity for the opponent to actually fight her without it.

3- She would have to choose between pushing the wave and roaming OR poke lane opponent

This is the most important aspect, I think, of giving her mana. If she uses her Q a lot on her opponent, she will be unable to push the wave properly ( and she would go oom, eventually ) assuming, at least, that her opponent position himself not so close to the minion wave ( the same thing you do against Xerath, so he has to choose between trying to Q you or the minion wave)

4- This would decrease her sinergy with tons of magic pen early

Katarina does a lot damage, that is true, especially if she has + + . Since she doesn't have to care about mana regen , she can buy all this items early that gives her a lot of magic pen ( assuming 8 magicpen runes, this build gives her 58 of flat magic pen ). Early-mid game, she deals basically true damage if her target has less than 58 of magic resist, and close to true damage to anything with less than 80 MR.

Things to change incase she doesn't get mana as a resource : Her passive - I think her passive should ONLY gives her resets when SHE kills, not on assists. As her lore say, she kills everything by herself, so changing it would not only fit her thematically, but would make her more balanced. On assists, if you don't want to take it off completely, make it only 50% effective (7.5s CD decrease). I think changing her to kills only reset would make her more balanced. She would probably need buffs after, don't know, but I think it is a good way to start.

I love katarina as a champion, since I started playing this game, but after several nerfs to her counters that made her really bad for a time, she fine made a comeback. Her abilities base damage are really low, with great scaling on her Q ( assuming you proc the mark ) and her R , making her more of a mid-late game threat. I don't want to see her butchered, nor permabanned