Could Reloaded-era Neo have hit Morpheus in the training program?

You mean a Neo that realized his powers? Yes, with hilarious ease.

A side question is why Neo couldn't hit Morpheus the first time.

The answer is pretty easy, Morpheus was walking us through it the entire time they were in the construct.

Neo hadn't fully internalized the truth yet--once he did, he could start bending and breaking rules the way Morpheus had.

Considering he landed a hit during their first sparing match, but he pulled his punch, I would say yeah

Well of course, easily. Remember end of first movie, he defeats those agents with ease using one arm against agent smith.

Are you trolling us? 🤣

Basically in the dojo fight, Neo starts out the fight all excited that he's just had all these martial arts downloaded into his mind, and Morpheus goes somewhat easy on him to allow Neo to become confident in his abilities. But then after Morpheus knocks Neo down he says.

"Good, adaptation, improvisation. But your weakness is not your technique." The point of this quote was that Neo's martial arts usage was technically flawless- but it wasn't enough. Later on after he's knocked Neo around a bunch he breaks it down for Neo.

"Do you believe that my being stronger or faster has anything to do with my muscles in this place?" The point of these quotes was that- Neo was still fighting as if he was in the real world- relying on the power of his muscles & letting feedback from his body (pain, tiredness, exhaustion) dictate his moves. Morpheus wants Neo to realize he's not fighting in the real world. He's not out actually out of breath because in reality he's just sitting in a chair. And in the same vein- theres no reason his speed & strength should be limited by how his muscles contract- because, again, Neo is really just sitting in a chair and not contracting any muscles. His speed is not dictated by his real life muscles- it is dictated only by what the program is telling Neo's brain his speed should be. 

"You're faster than this. Don't think you are, know you are." This and the quote at the beginning of the fight that the rules of the program can be bent or broken, is Morpheus trying to get Neo to let go of the information the program is feeding him about how fast, strong, or tired he is and override it within his mind by knowing that he is faster than the program tells him he can be. And only when Neo finally does that does he get fast enough to finally land a hit on Morpheus. 

Which all leads us to the initial question you asked- could Reloaded era Neo land a hit on Morpheus, and the answer is that he could most certainly. Within the Matrix, Morpheus & his crew use their knowledge of the Matrix's virtual reality to increase their speed, strength, durability to way beyond that of an average joe still plugged in. That's how they're all able to make all those crazy jumps and go up against squads of law enforcement on their own, but they're not able to fully override the Matrix's reality. When Neo becomes the One at the end of the first Matrix- Neo's mind becomes strong enough to not just bend the rules of the Matrix to become stronger, he can outright fully manipulate that reality. He becomes by far the fastest combatant of basically anyone else, including Morpheus. A simple illustration is how within the Matrix Reloaded, Neo beats up 3 Agents on his own without really any effort, and later on in the movie, Morpheus barely survives going up against just one.

Every redpill can break the rules of the matrix to a certain point. Thats why they can run on walls and shit.

I mean, Neo tanks multiple agents at once where everyone else struggles. It’s pretty obvious.

But Neo could have hit Morpheus in the first film already, so why is this is even a question? He only stopped from hitting Morpheus because he didn't want to. "I know what you are trying to do".

Real power is being capable of doing a thing and being able to not

Morpheus silently farts.

Morpheus: "You think its air that you breathe?"

Neo gets lung cancer.