"God" is capitalized as a proper name. You seem to use lower-case for many proper names. This is not the same as "a god" but refers to the entity that believers in a monotheistic religion call by this name.
If you want God to do something for you, there are many things you can say, but your second suggestion is not a natural one and your first one sounds like you want God to do something bad to someone else. Not a great thing to ask God to do for you.
Also, remember that you should punctuate your question as follows:
Do we need to say "May god teach you a lesson" or "god, may his work gets completed in time".
Student or Learner