Hi! I now have the official answer from West Sussex Highways:

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Enquiry ref: 17713985 - Hastings Primary Road Network

Thank you for contacting us about the classification of Havelock Road and Harold Place, Hastings.

Havelock Road & Harold Place are part of the A21 but do not form part of the Primary Road Network (PRN) and therefore the A21 and A259 PRN do not meet.
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Given the above, my intention is to remove highway=trunk from Havelock Road and Harold Place. Since I don't have a definitive statement about Cornwallis Terrace, I plan on leaving that as highway=trunk.