Does every point of secondary wavefront created from wavefront acts as a source of another secondary wavefront?

Huygens' principle states every point on the current wavefront acts as a source of secondary spherical waves. These secondary waves propagate outwards, in the forward direction, and a common tangent (an envelope) to all these waves constitute the new wavefront..but do these secondary wavefronts act as a source of another secondary waves?

Sure, a wave $$u(t,x,y,z)$$ propagates along both time and space. For every time slice, there is a wavefront and it's the source for the next wavefront in the next time slice.