Can a character jump vertically (with range 3 + strength modifier) to reach a flying enemy?
Or, is it impossible to combine a vertical jump with an attack?
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You can jump as part of movement, without even making a check if it's within your defined jump distance; and you can make an Attack action at any point during your movement. You can even split up your movement between the attack rolls you get, if you get more than one as part of an Attack action.
Player's Handbook, p. 190:
BREAKING UP YOUR MOVE
You can break up your movement on your turn, using some of your speed before and after your action. [...]
MOVING BETWEEN ATTACKS
If you take an action that includes more than one weapon attack, you can break up your movement even further by moving between those attacks.
Based on that, there's nothing in the rules that stops you from taking an attack action while in the middle of a jump -- or even making multiple attacks, taking a bonus-action attack, and Action Surging to use a second Attack action while still in mid-jump.
That said, hovering in the air while you attack over and over seems a bit silly to me (reminds me of Devil May Cry or Final Fantasy VII), so as a DM, I'd probably rule that you can only make one attack roll in that way before you come back down. In theory you could just jump again for each attack (spending movement to do so), but that still sounds iffy from a narrative perspective, at least to me, so if I decided to let you jump again to try a second swing (or third), I'd probably say it counts as a standing start -- but that's just my opinion.
On your turn, you can move a distance up to your speed.
Your movement can include jumping, climbing, and swimming.
See Basic Rules, Movement and Position
Reach is a specific game term, reach determines your range without movement, it also affects your opportunity attacks. For player characters their reach is 5 ft, unless you have a weapon with the Reach property.
There's no raw way to attack a flying enemy more than five feet above you when you have a 5-foot reach.
Increase your reach; reach weapons, having a naturally longer reach, lunging strike from the Martial Adept feat, etc.
Ready an attack against melee attacks. If the flyer has to come down to attack, and then flies away again, a melee attacker can go to the attack location and ready an action to attack when the enemy comes into range.
For the sake of completeness: use a ranged attack. You can pull out that handaxe or a bow or whatever.
We've had to deal with this a few times at my table and we allow what you're asking about. In the scenario where an attacker is flying ten feet above a melee attacker with an attack range of 5 feet, they can spend 10 feet of movement to attack the enemy above them (5 feet up and 5 feet back down).
While everyone has seemed satisfied with that, if it were to be a frequent occurrence (i.e. - if I were to start purposefully planning encounters that relied on flying enemies and reach), I would probably adjust the rules a bit in the favor of the enemies.