I'm sorry to hear you're experiencing the same.
I just got off the phone (as in regular phone call) with someone, and thankfully the audio on my end (going out) and their end (coming in) we're both flawless.
And this is interesting.. as I write this, I am finding that all keyboard presses are registering the appropriate sounds without any lag or skipping. Additionally, I am getting normal responsiveness with the audio volume controls.
Let me try my podcast app...
Wow! My podcast app (iCatcher) is also working properly again.
I wonder if somehow receiving a phone call (again - normal phone app feature, not FaceTime or Skype or Zoom) somehow caused whatever bit of code in the iOS update to enter the proper state of function or self-correct?
If you're reading this and having the same audio glitches and UI lag issues, try making a regular phone call and see if that corrects anything. If not, ask someone to phone you, and see if receiving a call might fix the issue.
Or perhaps updating to iOS 13.4.1 simply caused the phone to run some level of background maintenance that was a bit too aggressive, causing the audio & UI glitches, and it simply completed those background processes around the time I received the phone call that seems to have magically fixed the problem for me?
Hoping my phone behaves again, and that what I've shared might be helpful to anyone else with the same issue.