Katerina, as much as....means.... however much, or no matter how much automatic cognitive processes....thought processes that we do not control completely (unconscious or sub-conscious processes) largely influenced.....to a large extent influenced our responses to life experiences....what we think, feel, or do To paraphrase: No matter how much we like to think we can be in control of our judgments and behavior, in fact thought processes over which we do not have complete control influence to a large extent what we think, feel and do.

"as we like to think we can..." означает, что нам нравится думать, что мы можем контролировать ... бла бла бла. Можем ли мы это на самом деле, остаётся под большим вопросом, но несмотря на это придерживаться этого мнения нам очень нравится. "the evidence is growing" означает, что появляется всё больше и больше свидетельств этому.

Я бы перевела как "как бы нам не нравилось/хотелось думать, что".

As Uleman alluded to (alluded means suggested or hinted at), The evidence is growing (there is more and more evidence) that, as much as we like to think we can be in control of our judgement and behavior (we assume that on the surface we can decide how to judge or react to events or situations), in fact we are largely influenced by automatic cognitive processes that shape our responses to life experience (subconscious established attitudes may dictate how we react to external stimuli and this could be called "triggered response"). Hope this helps.