Your search for "Letters maintaining relationships: thanks, apologies, congratulations and best wishes" returned 10 letters.
Sir George Pierrepont writes to thank Bess (Lady St. Loe) for wine received from her husband, William St. Loe, and for the 'bountious goodness' shown unto his son Henry at London. He apologises that he is not able to travel to Chatsworth due to illness but reiterates his liking for the match between his son Henry and Bess's daughter Frances. Adds some suggestions for arranging the marriage settlement.
Elizabeth Cavendish (later countess of Lennox) writes to her mother, Bess (countess of Shrewsbury), to thank her for lending her a litter for a journey.
Elizabeth Smyth writes to Bess (countess of Shrewsbury), thanking her for accepting 'of part of my little one' (Bess stood as godmother to Smyth's daughter), and for a gift, 'a bowl with a cover'.
Edwin Sandys, archbishop of York, writes to thank Bess (countess of Shrewsbury) for her 'courteous letters', a 'fat stag' and an 'honorable offer', the latter of which he does not answer for reasons entrusted to the bearer of the letter.
Sir John Manners writes to Bess (dowager countess of Shrewsbury) to thank her for her good wishes in relation to the birth of his grandchild (John, later eighth earl of Rutland), mentioning that his son (George Manners) is awaiting 'to know your pleasure about the christening'.
Roger Manners, earl of Rutland, writes to Bess (dowager countess of Shrewsbury), thanking her for kindness shown to his cousin, Edward Talbot (also Bess's stepson), on a recent visit.
Thomas Howard, earl of Arundel, writes to his grandmother-in-law, Bess (dowager countess of Shrewsbury), expressing thanks and 'obligation' for favour shown unto him.
Thomas Howard, earl of Arundel, writes to his grandmother-in-law, Bess (dowager countess of Shrewsbury), regarding his newborn son: Bess had previously been asked to stand as godmother, but now the queen has requested to be the child's godmother.
Mary, countess of Shrewsbury, writes to her mother, Bess (dowager countess of Shrewsbury), thanking her for a gift of ermine fur and reporting news related to the christening of Bess's great-grandson James (Howard), including the queen's journey to attend it. She is glad that Bess's hip pain has lessened, although she herself is troubled with a pain in her head.
William Kniveton writes to Bess (dowager countess of Shrewsbury) to send elaborate thanks for her good will and favours, with best wishes for the new year.