M-dwarfs in extremely wide binary systems are very rare, and may thus have different formation processes from those found as single stars or close binaries in the field. In this paper we search for close companions to a new sample of 36 extremely wide M-dwarf binaries, covering a spectral type range of M1 to M5 and a separation range of 600 - 6500 AU. We discover 10 new triple systems and one new quadruple system. We carefully account for selection effects including proper motion, magnitude limits, the detection of close binaries in the SDSS, and other sample biases. The bias-corrected total high-order-multiple fraction is 45% (+18%/-16%) and the bias-corrected incidence of quadruple systems is < 5%, both statistically compatible with that found for the more common close M-dwarf multiple systems. Almost all the detected companions have similar masses to their primaries, although two very low mass companions, including a candidate brown dwarf, are found at relatively large separations. We find that the close-binary separation distribution is strongly peaked towards < 30AU separations. There is marginally significant evidence for a change in high-order M-dwarf multiplicity with binding energy and total mass. We also find 2-sigma evidence of an unexpected increased high-order-multiple fraction for the widest targets in our survey, with a high-order-multiple fraction of 21% (+17%/-7%) for systems with separations up to 2000AU, compared to 77% (+9%/-22%) for systems with separations > 4000AU. These results suggest that the very widest M-dwarf binary systems need higher masses to form or to survive.
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