I have a problem with the following sentence:
She usually works at home on a Thursday.
Does it mean she works from home one Thursday every month?
By itself, the sentence is a little vague, and it could probably be interpreted a couple different ways. With a little more context, though, we can make the meaning more clear:
This means she generally chooses to work from home on Thursdays, but she's allowed some flexibility. (Next week, for example, she might decide to work from home on Wednesdays instead.) As I read the bolded sentence in that example, my mind focuses on the phrase on a Thursday.
Let's reframe the sentence, though:
The speaker is telling someone why Leslie isn't in the office right now: because it's Thursday, and, on most Thursdays, Leslie works at home, not in the office. This time, as I read an identical sentence, but I put more emphasis on the at home part, because this second paragraph is focusing more on where Leslie currently is, whereas the first was focusing on which day she elects to work from home.
Both of those examples could have used on Thursdays instead of on a Thursday, but either one is acceptable. (Personally, I think the "on a Thursday" versions sound a bit more colloquial or homey.)
J.R. has already given an excellent answer, but I'd like to add that outside of this context, "On Thursday" can often mean "This coming Thursday" but not necessarily any Thursday after that.