The correct answer is option 'A': Battle of Talikota.

The Battle of Talikota, also known as the Battle of Rakshasa-Tangadi, took place in 1565 between the Vijayanagar Empire and the Deccan Sultanates. This battle had a significant impact on the Vijayanagar kingdom and weakened its power and influence in South India.

Below are the reasons why the Battle of Talikota weakened the Vijayanagar kingdom:

1. **Causes of the Battle**: The Vijayanagar Empire was engaged in conflicts with the Deccan Sultanates for years, as both sides sought control over the lucrative trade routes and territories. The battle was a culmination of these tensions and power struggles between the two sides.

2. **Alliance against Vijayanagar**: The Deccan Sultanates, including the Sultanates of Bijapur, Golconda, Ahmadnagar, and Bidar, formed a grand alliance against the Vijayanagar Empire. This alliance united the forces of the Deccan Sultanates against Vijayanagar, posing a significant threat to its power.

3. **Military Strategies**: The Deccan Sultanates adopted effective military strategies during the Battle of Talikota. They strategically lured the Vijayanagar forces into a trap by pretending to retreat, thereby isolating and encircling them. This led to a chaotic situation for the Vijayanagar army and weakened their defense.

4. **Superiority in Numbers**: The combined forces of the Deccan Sultanates outnumbered the Vijayanagar army. This numerical advantage played a crucial role in the outcome of the battle. The Vijayanagar forces were overwhelmed by the sheer size of the enemy army, making it difficult for them to hold their ground.

5. **Leadership Crisis**: The Vijayanagar Empire faced a leadership crisis during the battle. Rama Raya, the regent of Vijayanagar, made several strategic errors and failed to provide effective leadership. This further weakened the morale and effectiveness of the Vijayanagar army.

6. **Destruction and Losses**: The Battle of Talikota resulted in the complete defeat of the Vijayanagar Empire. The capital city of Vijayanagar was plundered, looted, and destroyed by the victorious Deccan Sultanates. The city never fully recovered from this devastation, leading to a decline in its power and influence.

In conclusion, the Battle of Talikota significantly weakened the Vijayanagar kingdom due to a combination of factors such as the grand alliance against them, effective military strategies employed by the Deccan Sultanates, numerical superiority, leadership crisis, and the destruction caused during and after the battle. This event marked a turning point in the history of the Vijayanagar Empire and led to its eventual decline.