There being 問題

你的問題是關乎分詞子句，分詞子句的主詞，一般與主要句子的主詞是一致，如：

Walking along the street, I met a fried.
Having read the book, I could answer the question.

Being ill, I did not go there

但若分詞子句的主詞，並不是主句的主詞時，那時，我們就必須說出分詞子句的主詞，否則就會出現dangling participle，如：

Being fine, I went out for a walk.若果這句的fine是形容天氣，而不是形容I，則這句的句子的意思，就表達錯誤，若是fine是形容天氣，這句應寫成：

The weather being fine, I went out for a walk.

分詞子句的主詞，與主句的主詞不同的例子很多，如下：

The rain having ruined my hat, I had to buy a new one.
There being no taxi, we had to walk.
It being very stormy, she stayed at home.

你的句子出現There，其理由也是如此


希望幫到你！

The sentence is correct. It is rather formal.
"There" behaves like a subject in the -ing clause.

An example from a grammar textbook:~

There being no further business, the meeting was concluded.

Human being as subject=I was no banana left.However,
The subject of phrase be a pronoun=there,eg:-
There is no banana left(in the house),is there?
If the formality there is used,verb agrees with the subject,:-
There are no bananas (plural in nos.)
There is no banana (singular in no.)
In conversation the participle phrase is usually made into an Adverb Clause,eg:-
Chnage "There being " into:-
As there was no banana left,I went...
As there being no banana left, I went.....
Hence care should be taken in using the (mis-related participles) construction to make sure that the participles are correctly related. The word to which the participle relates should be the same as the subject of the verb.namely,the participles should be correctly related.

Yes, the sentence is corect.
The word 'there' serves as the subject.
No, the word 'there' cannot be removed from the sentence. Without the word 'there', 'I' would become the subject. The sentence would mean something like "I went to the supermarket to buy some bananas because I was no bananas left" instead of "I went to the supermarket to buy some bananas because there was no bananas left'.