If Susan Lambert had married Richard Goodyear anytime after the birth of their child, Harriet's Lambert surname would have been dropped and she would have used Goodyear as her surname.
Richard or a member of his family may have voluntarily paid an allowance towards Harriet's upkeep if Richard acknowledge paternity. If he didn't either cough-up or offer to marry her, the churchwardens and poor law guardians would have been on to him to persuade him to do the latter or make him do the former to prevent Susan and her child becoming a burden on the parish. Susan may have voluntarily named Richard as father. If she didn't, pressure would have been applied to make her name the father. Look up Susan's and Richard's names in local quarter session petitions and bastardy orders for an affiliation and maintenance order.
Was Susan born in the parish or had she gained residence status by being employed there continuously for a year? If neither, the parish authorities would have tried to send her back to where she belonged; in that case there might be a settlement order, followed by an appeal. Look for a settlement order with her name on in quarter session petitions.
2 of my 3xgreat-grandfathers were a bit naughty as youths. One was served with an affiliation & maintenance order in 1823. He was named as father in the baptism register and in the burial register when his son died, aged 7. The other had a son with his future wife in 1841. The father's name was on the birth certificate and the baby was given the forename of his paternal grandfather. However there was no mention of a father in baptism register or in burial register 6 months later.