What is a specific, but limited, time (usually in children's lives) during which the child is particularly susceptible to environmental influences relating to some specific facet of development?

Sensitive Period.

Explanation:

Some researchers in developmental psychology and biology, consider the critical period in a child's development to consist of two sub-periods: 'strong critical period' and 'weak critical period'.

The latter is referred to as a sensitive period. During this period child learns such skills as grammar processing, vocabulary acquisition, and sports training. Skills acquired during this period can be also further improved at later stages of development.