If you do something in a grudging manner (or grudgingly), you do it with reluctance. Doing homework, paying taxes, and saying sorry are all commonly done in a grudging manner.
"Grudging apology" and "grudging acceptance" are probably the two most common uses of this word, which perhaps says something about our lack of grace as a species. From the medieval French word groucher, meaning "to murmur or mumble." A related word is grudge, meaning a long-lasting sense of ill will dating to a previous incident.