[Abridged] Classical novae (CNe) represent the major class of supersoft X-ray sources (SSSs) in the central region of our neighbouring galaxy M31. We performed a dedicated monitoring of the M31 central region, aimed to detect SSS counterparts of CNe, with XMM-Newton and Chandra between Nov and Mar of the years 2009/10, 2010/11 and 2011/12. In total we detected 24 novae in X-rays. Seven of these sources were known from previous observations, including the M31 nova with the longest SSS phase, M31N~1996-08b, which was found to fade below our X-ray detection limit 13.8 yr after outburst. Of the new discoveries several novae exhibit significant variability in their short-term X-ray light curves with one object showing a suspected period of about 1.3 h. We studied the SSS state of the most recent outburst of a recurrent nova which had previously shown the shortest time ever observed between two outbursts (about 5 yr). The total number of M31 novae with X-ray counterpart was increased to 79 and we subjected this extended catalogue to detailed statistical studies. Four previously indicated correlations between optical and X-ray parameters could be confirmed and improved. We found indications that the multi-dimensional parameter space of nova properties might be dominated by a single physical parameter. We discuss evidence for a different X-ray behaviour of novae in the M31 bulge and disk. Exploration of the multi-wavelength parameter space of optical and X-ray measurements is shown to be a powerful tool for examining properties of extragalactic nova populations. While there are hints that the different stellar populations of M31 (bulge vs disk) produce dissimilar nova outbursts, there is also growing evidence that the overall behaviour of an average nova might be understood in surprisingly simple terms.
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