Another answer I wrote at Quora.
A2A. Unless you think that’s a weakness that really needs explaining or that’s a weakness that actually explains your motivation to do a PhD, I see no reason to state it in your motivation letter.
A weakness that really needs explaining is a weakness that is so glaring and important, that it alone would prevent you from getting an offer. So, you try to explain to try to have a chance. Even so, don’t write too much about it. It may sound like an excuse.
A weakness that explains your motivation could be a good thing in some cases, even it may be risky to do that in your motivation letter. In my case, I wasn’t able to overcome a problem I found during my professional career. Something like that may be considered a weakness, right? After all, it’s not a case of success, but failure.
But after that, I became very interested in learning about ways to solve that problem. In turns out that the problem I was trying to solve is a major issue for Marketing research. So, I had to explain a weakness I had in order to show how a PhD could help me to achieve my goals.
Even so, don’t waste much time with your weaknesses. If a school is interested in you and about your explanations for a weakness, they will probably do that during an interview. During my interviews, they asked questions about the weaknesses they perceived in my application profile. Even weaknesses I had not thought about.