An easy way to determine the equations of motion in a non-inertial reference frame is to apply the chain rule to the (relatively simple) equations of motion in an inertial reference frame.
In the inertial reference frame, the equation of motion for the system you describe is (probably)
\begin{align*}
\frac{d}{dt} \vec p = \frac{d}{dt}(m \dot{\vec{x}}) = \vec T
\end{align*}
The coordinates $\vec \xi$ of the non-inertial reference frame are related to those of the inertial frame $\vec x$ by the equation $\vec \xi + \frac{1}{2}\vec a t^2 + \vec v_0 t = \vec x$, assuming there is no rotation and that the relative acceleration and initial velocity are $\vec a$ and $\vec v_0$ respectively.
With the above assumptions, $\dot{\vec{x}} = \dot{\vec{\xi}} + \vec a t + \vec v_0$, and $\ddot{\vec{x}} = \ddot{\vec{\xi}} + \vec a$, and so
\begin{align*}
m\ddot{\vec{x}} = m(\ddot{\vec{\xi}} + \vec a) = \vec T,
\end{align*}
or
\begin{align*}
m\ddot{\vec\xi} = \vec T - m\vec a
\end{align*}
Here it would be natural to refer to $m\ddot{\vec{\xi}}$ as the "pseudo-force" because it includes fictitious forces associated with the nonzero acceleration. In a rotating reference frame, such as that of any observer stationary with respect to the surface of the Earth, these fictitious forces also depend on the pseudo-velocity (somewhat like the velocity dependent forces that come up in electromagnetism), and can be used to infer one's distance to the axis of rotation.
