  Optical lattice experiments which probe the effect of disorder on
superfluidity often use a speckle pattern for generating the disorder. Such
speckle disorder is spatially correlated. While fermionic superfluidity in the
presence of uncorrelated disorder is well studied, the impact of correlated
disorder, particularly on thermal properties of the superfluid, is poorly
understood. We provide a detailed study of the impact of speckle disorder, for
varying speckle size and disorder magnitude, on the ground state and thermal
properties of a Fermi superfluid. We work in the strong coupling regime of
BCS-BEC crossover in a two dimensional lattice. For a fixed disorder strength,
an increase in speckle size leads to smoothening of the self-consistent
background potential, increase in the critical disorder needed for a
superfluid-insulator transition, and an increase in superfluid $T_c$ . Along
with these hints at decrease in effective disorder, speckle correlations also
suppress the superfluid gap and the gap formation temperature - effects
normally associated with increasing disorder. We correlate these effects with
the effective potential and the single particle localisation effects in the
ground state
