Particle Goal Question

Hello,

I am very new to this and I have an animated polygon mesh and some spherical particles which are attracted to the mesh. At the moment, the particles are attracted to the mesh in a rather uniform way, forming the shape of the mesh in layers.

I am wondering if someone could help me to control the particles so that one by one they are attracted to goal but also land on a random part of the mesh.

The desired effect is that the spherical particles will one by one appear on the mesh in a random location until they have covered it.

Thank you!