The most appropriate word for use in the example sentence is 'absenteeism', although I would argue that, considering the meaning of 'absenteeism', 'absence' would not be inherently wrong (with the sidenote that 'absenteeism' is perhaps less..ambiguous).
That said, 'absence' indicates a person's non-presence, there is no plural form of 'absence' to describe the fact that he, a single individual, is away, as far as I have been able to determine.
Wikipedia also holds some good information on the word and its usage.
Looking at 'absence' here, under Examples from the Web for absences, context appears to clearly showsshow 'absences' being used to refer to the number of times or instances non-presence was observed.
In conclusion, 'absences' can be used with 'frequent', but it creates some ambiguity as to what exactly is being referred to (his absence, or the number of times he was absent), and leaving open the question of what his wife is trying to cope with exactly (his absence, or the number of times he is away).