When Beckett enters Castle's study, she's holding two cups. When the shot cuts to show her from behind, she's only holding one. When the shot cuts back, she's holding two cups again.
In the break room at the precinct near the end, Beckett accepts a cup of coffee from Castle. As she takes it, she holds it in both hands, but when the shot cuts, she's only holding it in her right hand.
Anyone born and brought up in a region of England may speak English with an accent or use a local dialect with local words included. They would not, as a native English speaker, want or need to attend a class for English as a second language as a character is portrayed here.