I paid $3900 to a plumber in 2019 to replace my broken sewer lateral. Through a grant program the local water department reimbursed me $1950 of that cost two months after the work was performed. Do I need to include the reimbursed money under miscellaneous income?
No. However, it reduces your investment in your home (your cost basis).
When you sell your home, your capital gain is determined as
selling price minus cost basis.
Your cost basis is what you originally paid, plus the cost of any permanent improvements to the real property (real property is land plus any permanently attached structures). In this case, you would include $1950 as the cost of the improvement rather than the full price. If you owe capital gains tax later when you sell, you will owe a bit more than if you hadn't been reimbursed. (But most homeowners don't pay capital gains tax when they sell their personal home anyway.)
This doesn't sound like income. It sounds like the water department determined a portion of the plumber fee was their responsibility, and so reimbursed your expense. Alternatively, this may be a rebate as an incentive to upgrade your sewer attachment. In either case, it's not income.