Final answer:
An integration hairpiece is designed to allow the client's hair to be pulled through for blending, which is ideal for those with thinning hair.
Step-by-step explanation:
The hairpiece that allows a client's hair to be pulled through for blending purposes is known as an integration hairpiece. Unlike other hairpieces or extensions, an integration hairpiece is designed with an open weft structure that allows for the client's natural hair to be seamlessly integrated with the piece. This method is particularly useful for individuals with thinning hair, as it can create a fuller look without the need for adhesives or tighter weaves that may cause stress on the natural hair.