There are some relevant entries in the settlement examination collection -- here is the one for Martha (Bethnal Green 21 Aug 1848):
13 White Bear Gardens aged 70 w[ife] of Joshua Clark deceased 15 years as to her son James Clark, see his wife’s examination fo. 264.
Married at St Mary Aldermanbury about 50 years (no cert) maiden surname Brooks
Hath by him 4 children: John 40, Mary 39, James 36, Henry 34, Daniel 32.
Husband rented a house in a court of 2 houses in Old Nichol St at 4/- a week when John was born & held same above a year & then moved to No 1 Hudson’s Court opposite Bazin House Kingsland Road at 4/- a week & held same 20 years & 3 months. Mr Day [?] landlord for several years & never etc.
There appear to be a few closely related Clark families in this area of Shoreditch from the 1830s. I'm related to a Henry Clark born around 1820, married Caroline Paget in 1844, father *John* a sawyer. His addresses include 21, 11 or 12 White Bear Gardens, Harris' Buildings, then 2 Mail Coach Yard from the early 1850s.
Interestingly, this Henry is the witness for Martha's death even though he does not appear to be her son (Death certificate 4 Jul 1854: Martha Clarke, female, 77 years, widow of Joshua Clarke sawyer, phthisis 1 year, the mark of Henry Clarke present at the death 2 Mail Coach Yard Kingsland Road Shoreditch)
It's plausible that this Henry is the son of Martha and Joshua and his marriage certificate mixed up 'John' and 'Joshua'. If so, he would be the brother of your James. However, I have not been able to trace back further than Henry with any certainty.
There are other Clarkes at 1 and 8 White Bear Gardens and in Harris' Buildings (which was adjacent). It's very complicated so proceed with extreme caution!
best,
Mike
