  In this paper, we establish second order estimates for a general class of
fully nonlinear equations with linear gradient terms on compact almost
Hermitian manifolds. As an application, we first prove the existence of
solutions for the Monge-Amp\`ere equation with linear gradient terms for
$(n-1)$-plurisubharmonic functions, originated from Gaudochon conjecture, in
the almost Hermitian setting. Second, we solve the Monge-Amp\`ere equation and
Hessian equations with linear gradient terms. Third, we give the $C^{\infty}$ a
priori estimates for the deformed Hermitian-Yang-Mills equation with
supercritical phase. At last, we prove the existence of deformed
Hermitian-Yang-Mills equation and complex Hessian quotient equations under
supersolutions.
