Hi,
I think this might be Jane's info. the first bit you already have but I'll put it here anyway so that it makes sense to me

Jane BURFIELD
Eastbourne, Sussex
Birth Date: 23 Feb 1820
Baptism: 23 Apr 1820
Daughter of James and Rebecca
Marriage2 Jul 1843
St. Clement Danes, Westminster
Robert AGUTTER
Father: John AGUTTER
Jane BURFIELD
Father: James BURFIELD
ChildrenBirthRobert James AGUTTER Jun Q 1844 St George Hanover Square 1 20
mmn: BURFIELD
Wesleyan Methodist, Sandown, Isle of Wight (Ryde Circuit)
Birth Date: 7 May 1844
Baptism Date: 18 Oct 1844
Robert James
Son of Robert and Jane AGUTTER
Abode: London
DeathRobert James AGUTTER Dec Q 1844 St George Hanover Square 1 19
Age in years: 7
(This age is from the GRO site, I have seen quite a few examples where under the column "Age at Death (in years)" a figure is put and it doesn't say months when it is. I think this is your Robert aged 7 months.)
BirthHenry George AGUTTER Dec Q 1845 St George Hanover Square 1 353
mmn: BURFIELD
BirthGeorge AGUTTER Mar Q 1848 Westminster 1 452
mmn: BURFIELD
I think Robert AGUTTER dies in 1848,
DeathRobert AGUTTER Dec Q 1848 Westminster 1 353
Age: 35
Westminster Burials
Robert AGUTTER
Abode: St Johns Westminster
December 17 1848
Age: 35
In 1851 Jane is on the Isle of Wight,
1851 HO107/1664 f.243 p.39 Newchurch, Isle of Wight
Castle Street, Ryde
Henry KEMP Head M 36 Vitualler Hants Ryde
Maria " Wife M 35 Guildford, Surrey
Emma " Daur Un 9 Scholar Hants Ryde
Henry " Son 7 Scholar " "
Caroline " Daur 5 Scholar " "
George " Son 2 Scholar " "
Jane AGGATER Serv 28 Servant Sussex East Bourne
Ann PEACH Serv 23 Servant Hants Freshwater
In 1853 Jane she gets married,
Jane AGUTTER Dec Q 1853 Isle of Wight 2b 805
Charles PHILLIPS
1853
All Saints, Newchurch
Jane AGUTTER
Father: James BURFIELD
Charles PHILLIPS
Father: Jacob PHILLIPS
Jane and Charles can be found on the 1861 census with their children, then Jane is a widow on 1881 and 1891, she dies in 1899. Let me know if you need any help with finding this info

I will look for Henry George AGUTTER and George AGUTTER later on

Regards,
Daisy