Air carrier Captain reported an automation error between two similar sounding fixes; WITEN and WINEN; resulted in ATC issued course correction.

Narrative

After passing NEERO; the automation was set to fly toward WITEN when; in fact; it should have been WINEN. This was not caught at any point as WITEN and WINEN happen to be relatively close to one another. ATC gave an instruction; shortly after passing NEERO and on our way to WITEN; to 'Turn right heading 160 for military airspace.' We complied and thought nothing of it. ATC then asked what our filed routing was and this is when we caught our spelling mistake. ATC gave instructions soon after to fly direct to WINEN and continue the remainder of our filed routing.When preparing the flight for departure; I had two flight attendants asking several questions about the aircraft as they had not actually been on one yet nor had they flown for this major partner. I also had a maintenance discrepancy that I was on hold with Maintenance Control for. We were also on an about 2.5 hour delay for crew rest. I was still completing my pre-flight duties while waiting for maintenance to get back to me. This is when we checked the routing in the FMS and the misspelled fix was not caught by either of us. WITEN and WINEN are close enough to each other that the route did not look wrong in any way. After changing the routing back to WINEN in the air; the time only changed by a few minutes.This could have been avoided by slowing down and doing one major task at a time instead of trying to juggle maintenance; pre-flight duties; and helping other crew members.