Richard Percy Cocking married Caroline Sophia Tornquist on 15 Jun 1816 at St Luke, Chelsea.
According to this old post, she was born in Sweden:
https://www.ancestry.com/boards/surnames.tornquist/9/mb.ashxRichard and Caroline were the parents of Henry Percy William Cocking (1816-1840).
So yes it would all appear that Richard Percy Cocking's father Robert Cocking had a first wife Jane who he married presumably in Ireland and he was in Dublin in the 1790s.
Robert Cocking's second wife Anna Maria Cocking was born abt 1786 and died before 27 May 1876 aged 90 in the registration district of Edmonton. she was buried on 27 May 1876 at West Norwood Cemetery.
I have found her in the 1861 census in Newington as an independent with an annuity. She is in the 1871 census in Tottenham with her niece and great niece.
As Henry Percy William Cocking died in 1840 in Lambeth I had a look for other Cockings in the area.
To my surprise there were two Richard Percy Cockings in the Birth and Death registers.
The first was for a Richard Percy Cocking who was born in Westminster in 1838. The GRO index shows the mother's maiden name was Lloyd and that there was another child born to this unknown Cocking and unknown Lloyd in 1840 called Jane Percy Cocking.
I have not been able to find the parents names as yet. But there is this family in the 1841 census in Chelsea which might be them:
Sarah Cocking 30
Louisa Cocking 7
Fanny Cocking 4
Richard Cocking 2
Jane Cocking 7 Mo
The second Richard Percy Cocking was born in Lambeth 1847 and his mother's maiden name was Hobson.
With this information I was able to find the marriage for a Robert Cocking to a Mary Ann Hobson on 7 November 1844 at St Mary Lambeth. This marriage shows that his father was Richard Cocking (deceased) a stationer. Robert Cocking was a butcher of New Cut.
So it seems clear that the original Richard Percy Cocking who married Caroline Sophia Tornquist in 1816 had at least two sons. One being Henry Percy William (1816-1840) and the other being Robert Cocking who married in 1844.
This second Richard Percy Cocking died aged 14 months in 1848 in St Saviour Southwark.
In the 1841 census this Robert Cocking is in Streatham and is shown as a butcher. His age is given as 20 so a date of birth around 1821.
Looking at the GRO birth index with the mother's maiden names, after their marriage in 1844, Robert Cocking and Mary Ann Hobson had the following chidren:
COCKING, CAROLINE ANN mmn HOBSON
GRO Reference: 1845 D Quarter in SAINT GEORGE HANOVER SQUARE
COCKING, RICHARD PERCY mmn HOBSON
GRO Reference: 1847 D Quarter in LAMBETH Volume 04 Page 321
DIED 1848
COCKING, MARY ANN ELIZABETH mmn HOBSON
GRO Reference: 1850 M Quarter in LAMBETH Volume 04 Page 294
DIED 1854
COCKING, CHARLES ROBERT mmn HOBSON
GRO Reference: 1851 J Quarter in LAMBETH Volume 04 Page 292
COCKING, BENJAMIN JOHN mmn HOBSON
GRO Reference: 1853 M Quarter in LAMBETH Volume 01D Page 300
And that is all on the Cockings for now!