Accompanying circumstances. The first action is marked with 着 zhe indicating that it takes place at the same time as the second action:
Tā wò zhe wǒ de shǒu shuō：“xiè xiè 。”
Holding my hand, she said, ‘Thank you’.
(The first action, 握着我的手 wò zhe wǒ de shǒu ‘holding my hand’, takes place at the same time as the second action, 说 shuō ‘said’.)
Tā wēi xiào zhe huí dá wǒ de wèn tí 。
He smiled as he answered my questions. (lit. He smiling answered . . . )
Hái zǐ bào zhe wán jù xióng shuì zhe le 。
The child went to sleep holding the teddy bear in his/her arms.
Negative reinforcement. The second action is a negative expression reinforcing the meaning expressed by the first action, which is usually marked with 着 zhe:
Yé yé zuò zhe bú dòng 。
Grandpa sat completely still/motionless. (lit. Grandpa was sitting and did not move.)
(The first action, 坐着 zuò zhe ‘sat/sitting’, is reinforced by the second action, 不动 bù dòng ‘not moving’.)
Nǎi nǎi wò zhe wǒ de shǒu bú fàng 。
Grandma held my hand firmly in hers. (lit. Grandma was holding my hand and did not let go.)
 Yip, P. C., Rimmington, D., Xiaoming, Z., & Henson, R. (2009). Basic Chinese: a grammar and workbook. Taylor & Francis.