1. Identify the reagent

$p$-Toluenesulphonyl chloride is commonly called Hinsberg's reagent.

So, Option A is correct.

2. Its use in distinguishing amines

Hinsberg's reagent is used in the Hinsberg test to distinguish among primary, secondary, and tertiary amines.

In particular, it helps distinguish primary and secondary amines based on the solubility of the sulphonamide formed.

So, Option B is correct.

3. Reaction with primary and secondary amines

(i) Primary amine

A primary amine reacts with $p$-toluenesulphonyl chloride to form an $N$-substituted sulphonamide:

This product still contains one acidic hydrogen on nitrogen, so it dissolves in alkali by forming its sodium salt.

(ii) Secondary amine

A secondary amine reacts as:

The product has no hydrogen on nitrogen, so it is not acidic and therefore insoluble in alkali.

Thus, the statement:

"On treatment with secondary amine, it leads to a product, that is soluble in alkali"

is incorrect.

So, Option C is NOT correct.

4. Reaction with tertiary amines

Tertiary amines, $R_3N$, do not have an $N-H$ bond and hence do not form sulphonamides with Hinsberg's reagent under test conditions.

So, Option D is correct.

5. Final conclusion

The statement which is NOT correct is:

6. Comparison with stored correct answer

Stored correct answer: $C$

My derived answer: $C$

They match.