Your search for "Letters at Christmas and New Year" returned 9 letters.
Frances Pierrepont (née Cavendish) writes to her mother, Bess (countess of Shrewsbury), with New Year's wishes, enclosing linen and a drinking glass as gifts.
Elizabeth Wingfield writes to her half-sister, Bess (countess of Shrewsbury), describing how well the queen liked the garments sent by Bess and her husband George, sixth earl of Shrewsbury, and how it 'has caused her to give out such good speeches of my lord and your ladyship as I never heard of'. A postscript reports that all are well at Hackney.
Bess's husband, George, sixth earl of Shrewsbury, writes to Thomas Baldwin, his servant, with instructions for settling financial and business matters, including some with Bess (countess of Shrewsbury), who needs £100 for a new year's gift for the queen. In a postcript, Bess asks Baldwin for speedy payment, and that the money be given to her son, William Cavendish.
Elizabeth Wingfield writes to her half-sister, Bess (countess of Shrewsbury), that she has consulted with Lady Frances Cobham (lady of the bedchamber to Elizabeth I), who does not advise 'giving money' (presumably as a gift to the queen). Bess's grand-daughter, Arbella Stuart, is well and a tailor has been sent for to make her a new green velvet gown. Lady Cheke has 'had long talk with her Majesty lately of my Lord's [i.e. Bess's husband, George's] hard dealing and the queen gave many good words what she would do for your honour'.
Lady Dorothy Stafford reports that she has presented Bess (dowager countess of Shrewsbury) and her grand-daughter, Arbella Stuart's new year's gifts to the queen. The queen has said she will 'be careful' of (i.e. act favourably for) Arbella and has sent her a token, albeit not as 'good as her Ladyship deserveth'. Lady Stafford asks Bess to keep the contents of the letter secret.
Henry Cavendish writes to his mother, Bess (dowager countess of Shrewsbury), at Christmas, with wishes for her health and blessing.
Henry Cavendish writes to his mother, Bess (dowager countess of Shrewsbury), conveying his best wishes for the new year, together with a New Year's gift from he and his wife, Grace (née Talbot).
Mary, countess of Shrewsbury, writes to her mother, Bess (dowager countess of Shrewsbury), to send New Year's greetings and a cushion for Bess to use in daily prayer.
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.