Here is the original item:
To get on his mother's good side,
(A) Rufus
offered to unpack the groceries,
(B) but dropped
the carton of eggs that Mom would later
need (C)
to make chocolate-broccoli muffins for the church bake sale.
You wanted to do this:
To get on his mother's good side Rufus
offered to unpack the groceries, but dropped
the carton of eggs that Mom would later need to make chocolate-broccoli
muffins for the church bake sale.
The comma after side is necessary. To get on his mother's good side is an infinitive phrase. When a phrase of this type introduces a main clause, you need a comma to attach it.
You might want to review the rules.