Тема: Complex Object (Сложное дополнение)
I saw you swaying to the rhythm of the music
Caught you playing, caught you praying to the voice inside you
"Swaying/playing/praying" are ing-forms of the verbs to sway/to play/to pray.
"Ing-forms" are used to speak about some processes, about something that is not finished. "Inf-form" can be the part of the Present Continuous: I'm listening to the news now/at this moment.
In the sentence "I saw you swaying" the phrase "you" +"swaying" completes (дополняет) the sense of the verb "saw". I saw (that) you were swaying = At the moment I was looking at you, you were swaying = I was looking at you when you were in the process of swaying (you have begun swaying and you haven't finished that process).
I caught you playing/praying. The phases "you playing /you praying" make the sense of the verb "caught" more concrete = these phrase serve as Complex Object (в предложении они являются сложным дополнением. Почему дополнением? - Они дополняют, конкретизируют значение основного глагола saw/caught. Почему сложным? Они состоят из двух элементов: местоимение (как в нашем примере)/существительное + ing-форма (в данных предложениях, а вообще в качестве второго элемента могут употребляться: прилагательное, инфинитив, существительное, даже наречие или порядковое числительное). Такие конструкции используются после следующих глаголов: to catch, to feel/ to hear, to get, to keep, to leave, to like/dislike, to see, to stop, to watch, to remember, to forget:
I felt him looking at me now and again.
When he arrived he found me reading.
Just as I got to the end of the corridor, I heard my telephone ringing again.
He saw me watching him.
I remember they/their teasing me about my nose. = They teased me in the past, and I remember it (their teasing) now.