Asthore is an Anglicization of Irish a stór; it is a term of endearment that is most often translated as 'my dear' or 'my darling', but more literally would be 'you are a treasure'. A literal translation is a bit difficult, however, as English doesn't use the vocative particle, embodied in the Irish a. Attempts at literal translations include 'O, treasure' and 'my treasure' (totally wrong!); most dictionaries simply leave it at 'treasure'. A stór is a shortening of the longer phrase, a stór mo chroí ('treasure of my heart').
As per to Google Ngram viewer, Asthore came crashing into English c. 1830, and faded away over the following decades, with a district and final drop off c. 1900. The sudden popularity is somewhat mysterious, and may in fact be a artifact of the written works that have currently been scanned and uploaded. However, the 1830s saw a resurgence in popularity of the ballad Molly Asthore, and a number of plays and short stories that focused on Irish characters. A few other corruptions of Irish endearments entered English at this time, such as acushla (short for a chuisle mo chroí, 'pulse of my heart') and avourneen (from a mhuirnín, 'my love').
Asthore is not pronounced the way it sounds; it is generally pronounced "asther" or sometimes "aster".