The Putney Inn [1] listed on their website does not specifically mention offering a complimentary breakfast. However, some websites do mention Putney farmhouse with free breakfast which might be a different property altogether.
If you are set on finding a farmhouse with free breakfast, I would recommend looking beyond The Putney Inn. Here are some options I found through a quick web search:
One Fanthorpe Street, Bed and Breakfast Putney [2]
Hurlingham Bed & Breakfast [2]
Hickory Ridge House Bed and Breakfast Inn [3]