How did the Third Carnatic War affect the political fortunes of the French in India? What were the causes of the French failure in their struggle against the English Company?

The Third Carnatic War was fought between 1757 and 1763 and it concluded with Treaty of Paris in 1763, which returned Chandernagore and Pondicherry to France, and allowed the French to have trading posts in India but forbade French traders from administering them. The French agreed to support British client governments, thus ending French ambitions of an Indian empire and making the British the dominant foreign power in India. Thus:

• Third Carnatic war ended the military and political power of French in India
• The French company was pushed to a corner and was confined primarily to Pondicherry. 

Reasons for failure of French:
Superiority of British naval power helped them to bring Soldiers and resources from Europe very easily. The lack of naval strength of the French compared to that of the English was one of the decisive factors for the failure of the French in India. The trade carried by the British East India Company was far greater than the French company. The superiority in trade has naturally made the financial position of British was far better than that of French. English East India company was a private enterprise but French company was excessively controlled  by government and did not enjoy any autonomy.  The recall of Duplex in 1754 A.D was one of main reasons for failure of French.
The acquisition of Bengal after Battle of Plassey provided enormous resources to the British which helps them in Carnatic wars.  English had three important bases Madras, Bombay and Calcutta. In Contrast French had only one strong post at Pondicherry. British recognized the importance of Bengal, the resources of which were liberally used in the third Carnatic War. In contrast, French influence in Hyderabad did not yield them adequate benefits. French were focused at the same time on Continental expansion in Europe which divided their resources.