Option 1, Ajmer, is not the correct answer. The province of Ajmer was not lost, recovered, and lost again during the mentioned time period. Therefore, we can eliminate this option.

Option 2, Kashmir, is also not the correct answer. While the Mughal Empire did face conflicts and challenges in the region of Kashmir during the mentioned time period, it was not a province that was specifically lost, recovered, and lost again. Therefore, we can eliminate this option.

Option 3, Bengal, is not the correct answer either. Although the Mughal Empire faced challenges and rebellions in Bengal during this time, it was not a province that was lost, recovered, and lost again. Hence, we can eliminate this option.

Option 4, Kandahar, is the correct answer. During the mentioned time period, the province of Kandahar was indeed lost by the Mughal Empire in 1622 A.D., recovered in 1638 A.D., and finally lost again in 1649 A.D. This province was strategically significant and was a contested area between the Mughals, Safavids, and Persians. Ultimately, it fell out of Mughal control and was lost permanently.