“She stood there chatting with her friend, with her child playing beside her.”
I understand the underlined part above is called “with absolute construction”, which is used when the subject in the construction is different from the one in main clause.
By the way, when I see two sentences below in our local material, I think their construction is different from the one above, since they lack the nouns acting as an independent subject;
1) My mother welcomed me [with giving me a big hug].
2) She always jogs in the morning [with listening to music].
I wonder if the two sentences including ‘with’ above are grammatically correct or not. If not, I think they should be amended into a adverbial participle clause by deleting ‘with’ as below;
1-1) My mother welcomed me, [giving me a big hug].
2-2) She always jogs in the morning, [listening to music].
Your opinions would be much appreciated.