with my mother being ill, 属于“with+宾语+现在分词”的复合结构通常在句中作原因状语，请看下面例句。

备注：去掉介词 with 属于“独立主格结构”作状语。去掉 being, 属于“with+宾语+形容词”复合结构作状语。

(1)my mother being ill (独立主格结构)

(2)with my mother ill (with+宾语+形容词)

(3)with my mother being ill (with+宾语+现在分词)

以上结构都是正确的，只是表达形式和方式不同。

I won’t be able to go on holiday with my mother being ill.因为妈妈有病，我无法去度假。

He fell asleep with the lamp burning. 他没熄灯就睡着了。

with+宾语+宾补”的复合结构还有以下几种，供参考。

【1】with+宾语+形容词

He often sleeps with the windows open. 他常开着窗睡觉。

Don’t speak with your mouth full. 不要满嘴巴食物说话。

【2】with+宾语+副词

He stood before his teacher with his head down. 他低着头站在老师面前。

He was lying on the bed with all his clothes on. 他和衣躺在床上。

【3】with+宾语+介词短语

She said good-bye with tears in her eyes. 她含着眼泪说了声再见。

The man was asleep with his head on his arms. 这个人头枕着胳膊睡着了。

【4】with+宾语+现在分词

He fell asleep with the lamp burning. 他没熄灯就睡着了。

I won’t be able to go on holiday with my mother being ill.

因为妈妈有病，我无法去度假。

【5】with+宾语+过去分词

He sat there with his eyes closed. 他闭目坐在那儿。

All the afternoon he worked with the door locked. 整个下午他都锁着门在房里工作。

【6】with+宾语+动词不定式

I can’t go out with all these clothes to wash. 要洗这些衣服，我无法出去了。

With such good cadres to carry out the Party’s policy we feel safe.

有这样的好干部执行党的政策，我们感到放心。

【7】with+宾语+名词 

He died with his daughter yet a schoolgirl. 他去世时，女儿还是个小学生。

He left home with his wife a hopeless soul.他舍家而去,撇下了绝望的妻子

with my mother being ill 也可以直接用 with my mother ill 表示。