How to detect influence on behavior
I think you can approach it as a logistic regression problem, where you will have a on/off (1/0) feature called "exposure_to_image". And your goal will be to detect if that coefficient is statistically significantly different than 0. If yes, then the probability of eating the item was influenced by the exposure. 

As for the assumption that they would have eaten it anyway: 
That's the whole point of having the control group. So your dataset will have people where exposure_to_image=1, and people for who exposure_to_image=0. All of them will have some baseline probability of wanting to eat the cake. But that coefficient tells you by how much that baseline is affected.