According to the current edition of the Real Academia Dictionary, a manda is indeed a promise made to God, the Virgin or a saint, or a Last Will Testament.
A lot of people used to leave in their Will some money to pay for masses for the resting of their own soul and for masses for the resting of the souls in the Purgatory. The document you cite has a font very difficult to read but it seems to indicate that the mandas are somehow related souls (almas).
Currently a lot of living people still pay mandas for favors received (miracles), not with money but by visiting a shrine (sometimes walking with a big group of people for miles and miles) or by binging a gift and placing it next to some image. I've seen some statues of saints in Mexico City surrounded by printed theses brought by university graduates, statues of the Child Jesus surrounded by toys, statues of St. Charbel with colored ribbons in his open arms, in some churches they cover some walls with small paintings describing a favor, or they may have a board in the wall where people pin small Sacred Heart pendants.

Best regards,
Victoriano Navarro