In this work, we keep pushing K2 data to a high photometric precision, close to that of the Kepler main mission, using a point-spread function (PSF)-based, neighbour-subtraction technique, which also overcome the dilution effects in crowded environments. We analyse the open cluster M 44 (NGC 2632), observed during the K2 Campaign 5, and extract light curves of stars imaged on module 14, where most of the cluster lies. We present two candidate exoplanets hosted by cluster members and five by field stars. As a by-product of our investigation, we find 1680 eclipsing binaries and variable stars, 1071 of which are new discoveries. Among them, we report the presence of a heartbeat binary star. Together with this work, we release to the community a catalogue with the variable stars and the candidate exoplanets found, as well as all our raw and detrended light curves.