  We consider a two-dimensional massless Dirac operator $H$ in the presence of
a perturbed homogeneous magnetic field $B=B_0+b$ and a scalar electric
potential $V$. For $V\in L_{\rm loc}^p(\R^2)$, $p\in(2,\infty]$, and $b\in
L_{\rm loc}^q(\R^2)$, $q\in(1,\infty]$, both decaying at infinity, we show that
states in the discrete spectrum of $H$ are superexponentially localized. We
establish the existence of such states between the zeroth and the first Landau
level assuming that V=0. In addition, under the condition that $b$ is
rotationally symmetric and that $V$ satisfies certain analyticity condition on
the angular variable, we show that states belonging to the discrete spectrum of
$H$ are Gaussian-like localized.
