The Banjhi period is the period after the First Flush for Darjeeling, but before the Second Flush:
Tea is one of the only plants that is used only for the vegetative growth, not its fruit or flower. The tea plant is an evergreen small tree by habit, but has been trained as a shrub/bush through the periodic management practices like plucking and pruning.
After the winter dormancy the new growth is highly prized as the first tea of the season. Depending on the region it may become white tea or green tea. In Darjeeling it becomes its famous "first flush" tea. After the first spurt of growth from the plant it goes into a second period of dormancy. This occurs about a month after first flush starts and is called the Banjhi period. The plant produces a small bud, which doesn't grow to full size, as well as a leathery larger leaf which has little taste and is hard to process.
During this two-week period the plant must be plucked to get rid of these leaves. Often teas are made from these leaves, but the result is a low-quality tea, often produced for local consumption. The Banjhi plucking allows the plant to grow the small and extremely desirable second flush leaves.