How does a new polyp form, if it does not receive help from the others?
Does it cut the supply once it develops its own mouth? Or does it develop through photosynthesis of the mat, and not the face of the paly/zoa?
Think of it in terms of sexual vs asexual reproduction for clarification.
Sexual
Corals reproduce sexually by either internal or external fertilization.
Internally fertilized eggs are brooded by the polyp for days to weeks. Free-swimming larvae are released into the water and settle within hours.
Externally fertilized eggs develop while adrift. After a few days, fertilized eggs develop into free-swimming larvae. Larvae settle within hours to days.
However, some corals are hermaphroditic (having both male and female reproductive cells). Others are either male or female. Both sexes can occur in a colony, or a colony may consist of individuals of the same sex.
Synchronous spawning occurs in many corals. Polyps release eggs and sperm into the water at the same time. This spawning method disperses eggs over a larger area. Synchronous spawning depends on four factors: time of the year, water temperature, and tidal and lunar cycles.
Spawning is most successful when there is little variation between high and low tides. The less water movement over the reef, the better the chance that an egg will be fertilized.
Once the larva settles on a substrate, it develops into a polyp.
Asexual
Environmental disturbances may dislodge some polyps or portions of colonies from the parent colony and deposit them on another part of the reef.
Sometimes, newly developing coral colonies split and form separate colonies.
Often a polyp produced by sexual reproduction initiates growth of a colony asexually by budding. Budding occurs when a portion of the parent polyp pinches off to form a new individual. Budding enables the polyp to replicate itself several times and at the same time maintain tissue connections within the colony. Later, the same polyp may reproduce sexually.
So therefore to answer your question, be it sexual or asexual, once the larvae/bud forms it is on its own. It has all the cells capable of forming a new colony by meoisis and mitosis. It needs no further assistance from the mother polyp/colony.