  We investigate the second Hochschild cohomology of $
\overline{\mathcal{M}}_{g,n} $, the moduli stack of stable $n$-pointed genus
$g$ curves. For the cases when $g = 0$, we prove that the second Hochschild
cohomology vanishes if and only if $n \neq 5$. For the cases $g > 0$, assuming
a conjectural HKR isomorphism for smooth Deligne-Mumford stacks, we prove the
vanishing for all $( g, n )$ except possibly for the 8 cases $
  ( g, n ) = ( 4, 0 ),
  ( 3, 1 ), ( 3, 0 ),
  ( 2, 2 ), ( 2, 1 ), ( 2 , 0 ),
  ( 1, 3 ), ( 1, 2 ) $.
