If this relates to the William Burdett bc.1801-5 (his age varies on census) Brington, Hunts. the GRO birth index on their own site which now shows maiden names shows that the maiden name of Sarah for son Avery Burdett's birth in 1843 was Stokes.
If this is the right family they married earlier than your starting point of 1831!!
They had 7 children on 1841 census and FindMyPast shows that William Burditt married Sarah Stokes 15/11/1825 All Saints, Northampton (after banns). FreeReg also has the marriage with the added information that the grooms abode(at the time of the marriage) was Oundle.
According to census their children were born at Warmington, Northants. but can only find baptisms from dau. Frances 1833 through to son Avery in 1843. At least 3 older children: William bc.1827, Harriet bc.1829 and Louisa bc.1831.
William Burdett's baptism appears to be the William Birdit bp.27/6/1802 Leighton Bromswold, Huntingdonshire, son of William and Hannah.
William Burdit married Hannah Marshall 19/10/1801 Leighton Bromswold. After William's baptism there in 1802 they had at least 2 children at Brington i.e. John bp.24/1/1805 and Avery (transcribed as Aveby) bp.Aug.1806. William Burdett was bur.15/1/1807 at Brington so I would guess his widow Hannah probably married again.
Brington and Leighton Bromswold are within spitting distance of each other.
Annette