In the sentence,"God have mercy on us," why is 'have' used instead of 'has?'
'God have mercy on us' is the subjunctive, and is equivalent to 'May God have mercy on us'.
You are expressing a wish. This is a similar construction to 'Long live the King'.
'God has mercy' is indicative, you are making a statement about the nature of God.
The statement 'God have mercy on us' carries with it the possibility that God might not have mercy on us, just as 'Long live the King' contrasts with the possibility that the King might soon die.
May is omitted at the beginning of the sentence:
[May] God have mercy on us.
May is known for its use to express wishes:
FORMAL: used for expressing a hope or a wish (Macmillan)
This is a request or a prayer, not a statement that God has mercy.
Here is an explanation: Quora.
This sentence is grammatically correct, but it is not universally accepted as correct in all contexts. The use of "have" instead of "has" is an archaism, meaning it is an older or more traditional form of the language. In modern English, "has" is typically used in the third person singular. However, in religious or poetic contexts, the use of "have" may be used to convey a sense of reverence or awe.
Here is another explanation from the same site.
It is actually the subjunctive mood. The example of this rarely used mood is in the phrase ‘God Save the Queen’ . It does look like the imperative, but it isn’t. Look how close this sentence is that is in the imperative. “Officer, save my puppy”
The subjunctive is when the thought is a wish or a desire, or sometimes something contrary to fact. Not actually a command.