Let's use a simpler example to examine closely:
X needs Y more than you do.
This means X has a greater need for Y than you.
X needs Y more than what you need.
The what here sounds like it's trying to introduce an additional thing, let's call it Z. So this means X has a greater need for Y than the amount of need you have for Z.
So this should illustrate how the second sentence in your example seems to talking about a third random thing out of nowhere and is confusing.
You could say "X needs Y more than Y that you need" if you want to repeat Y for extra clarity. But this is not usual.
Also 'more than you do' ,'more than you need' and 'more than you do need', there all are grammatical and mean the same thing
When comparing two actions with than, the verb is typically the same on both sides of the than.
I walked farther than he walked.
Because the verb is the same, the second one can be omitted.
I walked farther than he.
If you want to use the emphatic mood, which is a form of to do + base word of verb, you can still do that.
I walked farther than he did walk.
But this really sounds awkward because usually only the do or did remains when this happens.
I walked farther than he did.
And that is the way it is usually said anyway, "I walked farther than he" has a "formal" flavor and I don't hear it in real speech to often.
Far more commonly would be "I walked farther than him" - this is fine because in this case than is a preposition - than can work either as a conjunction or preposition.
With the pronoun "you", there is a tendency to include the do/did afterward to make clear we are comparing two actions, needs, etc. and not a need to a person.
I need this vacation more than you. (I could be saying I need a vacation more than I need you, so to make it clear I don't mean that, I can say I need this vacation more than you do.)