From personal experience completing this quest:
Is there any way to figure out what I missed and what more I need to
complete the task?
No there is no way to find a list of what you have and have not registered for that mission. I finished it before getting to Kadara so there are definitely plenty of enemies to register for them.
Also do you have to scan the enemies to register the enemies?
Yes you have to scan enemies in order for them to get registered for the mission. Simply seeing them for the first time does not count towards the mission.
Some enemies bodies do not persist after their death for me to scan
them, and scanning in the middle of battle is too hard.
In my experience I just had to get the scanner out ASAP and spam the scan button until it registered because some enemies do disappear after death. Pretty much every uniquely named enemy counts towards this quest so it shouldn't take too long.
Another tactic I had was trying to let my squad members tank or distract enemies, so I could quickly scan the new enemies, especially when I realized they were one that would dissapear.
Note: There is not much of a reward and there is no continuation after so it's not time sensitive that you finish it. You will come across well more than 20 types of enemies (at least named ones) within the first 3 planets.
Examples: Each unique Remnant (Breacher, Observer), Roekkar (Raider, Sharpshooter), Kett (Cardinal, Apostle), and Creature (Eiroch, Adhi) will give you a tick towards that mission.