The sentence might be referring to the time around the menses or period - possibly just before, or just after. Is is an incorrect, but relatively common usage to refer to that time as the part of the cycle that is, hm, "on" or visible - "when on your menstrual cycle..." or something like that - to refer to when you are on that phase of the cycle. Really, it is more of a case of too much talking around terms and stacking evasions on top of each other, or something, since people are kinda squeamish around biology.
It is also possible that the sentence was attempting to refer to the blemishes appearing in relation to the cycle in general, rather than referencing a specific phase - the grammar would still be wrong, but in a different way. In this case, it could be trying to be deliberately imprecise about which times in the cycle are relevant to the outbreaks of blemishes - due to individual chemistry, individuals may find them during different parts of the cycle, but during the same parts of that cycle for each individual. If this is what the sentence was trying to say, it would be more correctly "particularly around certain times of your menstrual cycle" instead of "around the time".