I couldn't find any Goodyers of any spelling variant at Bury but I did find a William Goodhew baptised on 9 March 1777, s/o William and Ann.
I checked all the neighbouring parishes for any Goodyer variant for the period and found nothing apart from at Coates which is near Fittleworth which is in turn north of Bury by a couple of miles.
I found the following at Coates all children of Richard and Elizabeth Goodyer:
John bap ? Sep 1782
William bap 26 Dec 1784/5 (there are two years given in different transcriptions for the parish)
Henry bap 1 Feb 1787
Richard bap 18 Feb 1787
Elizabeth bap 6 Jun 1790
Clement bap 26 Jan 1794
Henry bap 4 Jan 1795
Anne bap 18 Mar 1798
Hannah bap 8 Feb 1801
Mary bap 1 Jul 1804
I suspect this William is the one who subsequently has children baptised at Bury.
I couldn't find a burial for the first Henry but there were only 41 burials at Coates between 1562 and 1897 and no Goodyers!
At Sutton there is a gap in the records between 1780 and 1793 so I couldn't find Elizabeth Christopher's baptism record.
The parishes I checked are: Amberley, Barlavington,Bignor, Bury, Coates, Coldwaltham, Fittleworth (records not transcribed), Hardham, Houghton, North Stoke, Rackham (no records), South Stoke and Sutton.