What did a medieval beggar woman earn?
Alms only, and giving to anyone fit to work was a crime.
A wooden bowl, a shawl, and the church porch on a cold morning.

Where the money came from
On 18 June 1349, in the middle of the plague, Edward III issued a law on work. An estimated 30 to 40 per cent of the people of England had died, and the workers left alive were in great demand. The law ordered every able man and woman under sixty, with no land or trade of their own, to serve whoever asked, at the wages paid before the plague. A woman who refused could be sent to gaol until someone guaranteed that she would serve. It said many able beggars refused to work, living in idleness, vice and sometimes theft. Anyone who gave them anything out of pity or as alms could be jailed. So a beggar woman fit to work could lawfully be given nothing, while a law of 1351 let a haymaker take up to a penny a day: the haymaker earned a penny a day more. For a bed she could turn to a hospital. These gave shelter, food and prayer to the sick, the poor and travellers, not only medicine, and most were closely tied to the Church. The law largely failed: workers went on winning higher wages. In 1388 a new law told those too old or too ill to work to stay in the town where they lived, and made each district responsible for its own poor, though it too was little enforced.
Sources: en.wikipedia.org · en.wikipedia.org 2 · en.wikipedia.org 3 · en.wikipedia.org 4
What a beggar woman's pay bought in 1350
a farthing loaf of bread: out of reach, with no pay at all
a dozen eggs: out of reach, with no pay at all
Every price of 1350, with its source, is on the prices page.