Will my failing grade affect my chances of getting into a competitive programme or greatly lower my chances of getting a scholarship?
Unfortunately yes, this is quite possible and, at least in certain contexts, rather likely. Most good graduate programs in mathematics (and presumably elsewhere) see very high grades in general from their applicants. One or two imperfect grades can be easily dismissed (still, better is better...) but a failing grade is a big red flag. In many (American, I don't know how it is elsewhere) graduate programs, the minimum GPA required to maintain good standing is 3.0. Thus a lackluster performance in a graduate course often results in something like a B/B+. There is a lot of variation here and many problems make a de facto distinction between lower level / core courses in which "grades count" and higher level / optional courses in which it may well be expected that everyone gets the highest possible grade more or less automatically. Either way, a failing grade in a graduate course looks especially bad.
The more courses you take, the more one can discount any one grade. But if as you say you will be applying for admission and/or scholarships with one out of four grades a failing grade: well, I'm sorry to say it, but that doesn't sound good at all.
What can you do? First I would look into the prospect of getting the grade changed (though of course it may not be possible and in certain circumstances it may not even be appropriate to ask). If that's not possible, the matter becomes how to explain the grade in a way which makes it minimally alarming to people who are evaluating your application. In this regard I have to be honest again and say that your given explanation is not a great one: there was one problem set that was worth 40% of your grade, you didn't turn it in on time, and there was no way for you and/or the course instructor to rectify the situation? Not good. Your perspective may well be that the course instructor was an extremely unreasonable, ungenerous individual. And you may be right, but that doesn't really fix the fact that you didn't meet the course requirements. Maybe your entire program is somehow off...which is still not great.
I would try to have at least one faculty recommendation letter explicitly address this situation and explain it in a way which somehow allays the above concerns.