Why am I paying two lots of tax at the end of the year?

From September, two sums may be taken monthly instead of one

We are non-residents with a French second home which we have been letting out for part of the year. From September last year, two sums were taken monthly instead of one. Why would this be?

Tax on the income of any one year is taken in two bites: 

Firstly during the year in which the income is earned (on an on-account payment basis of expected tax and social charges in the case of rental income as in your case) and then secondly as a top-up if there is a shortfall of dues to be paid.

There can also be a refund if too much tax has been paid. In September two things may have happened: 

Firstly, the collection of the remainder of any tax due on the previous year’s income will have started. If this is more than €300 it will be taken in four, monthly, amounts usually on the 25th of the month. 

Secondly, the regular monthly instalment amount is adjusted in light of your latest tax assessment to now be 1/12th of the amount due under this new assessment.

This is usually taken on the 15th of the month. So this could explain the two payments each month in the latter part of the year.