Arthur is semi-illegitimate, his father having already gone through a marriage ceremony before being forced to marry Mrs Clennam. Arthur is his father's child by that first marriage. His real mother was a singer who was befriended by Frederick Dorrit (Amy's uncle - James Fleet). When Mrs Clennam found out, she took the child, Arthur, as her own in return for the woman never seeing her husband again and promising never to reveal his little indiscretion.
Arthur's father, in gratitude for Frederick Dorrit's befriending the girl, left a legacy to Frederick's niece - Amy.
That help?