It is hard to know what to make of TD 4 right now. So far convection has not developed around the center of circulation and has reduced over the day today. However, the depression's circulation is unusually large and well defined at this time. Complicating matters there is a large semi-circle of convection to the south of the center of circulation that is actually causing the center of circulation to shift south as the southern convection lowers surface pressures in that area.
With that said, a southerly shift in the circulation center has two significant effects. 1) It will cause the storm to stay south and miss the steering currents that want to cause it to drift northwest and recurve. As a result we will likely see a much more westward track. This is likely happening as all of the model forecasts have been very incorrectly forecasting TD 4 to move northwest, but instead its motion has been westward with a southerly drift induced by reformation of the circulation center. 2) As long as the circulation stays to the south, the Coriolis effect (turning of the flow due to Earth's rotation) will be weaker. As a result, the convection will have a hard time inducing a cyclonic circulation, which allows a strong pressure gradient to be maintained. Such a pressure gradient feeds moist air to the center of circulation and is the direct cause of strong winds which in turn provide even more fuel for the storm. However, if the depression can strengthen in spite of a weaker Coriolis effect at low latitudes, it is likely to rapidly strengthen as it moves north. Such a spin-up of the storm is analogous to a figure skater pulling in its arms. Ok, not really, but I'll pretend.