Average velocity is defined as the ratio in change in position to change in time,
v[ave] = ∆x/∆t
which on its own doesn't have anything to do with acceleration.
If acceleration is constant, the average velocity is the literal average of the initial and final velocities,
v[ave] = (v[final] + v[initial]) / 2
If this constant acceleration has magnitude a, the final velocity can be expressed in terms of the initial velocity by
v[final] = v[initial] + a*t
and plugging this into the previous equation gives
v[ave] = (v[initial] + a*t + v[initial])/2
v[ave] = v[initial] + 1/2*a*t
If the body in consideration is initially at rest, then
v[ave] = 1/2*a*t
which might be the relation you're looking for. But bear in mind the conditions I've underlined.
If acceleration is not constant and changes over time, so that the acceleration is some function of time a(t), then you can determine the velocity function v(t) by using the fundamental theorem of calculus. You need to know a particular velocity for some time to completely characterize v(t), though. For example, if you're given the initial velocity v[initial] = v(0), then
[tex]\displaystyle v(t) = v(0) + \int_0^t a(u) \, du[/tex]
or if you know any other velocity for some time t₀ > 0,
[tex]\displaystyle v(t) = v(t_0) + \int_{t_0}^t a(u) \, du[/tex]
