There is a difference between these constructions, and there is only one possible answer to your question.
1. You can use either ['get' + object + past participle] or ['have' + object + past participle] when the person who is the subject takes an active part in what happens. For example, if you arrange for the barber to cut your hair, by asking him to do this and paying him afterwards, you can say:
"I got my hair cut."
"I had my hair cut."
We often use this for services that we pay for e.g. 'We got the car repaired' or 'They had their house valued.'
2. However, if the event is something which happens against the subject's will or without the subject's consent or knowledge, then you can only use 'have'
This is the case with your example. The only possibility is 'had':
"She had her purse on the underground last week."
In this situation, the subject did not actively do anything - she was the victim of something that happened to her. This is why you had the answer 'got' marked wrong. Does that make sense?
Now, look again at the second sentence in the paragraph above. I wrote "you had xxx marked wrong". This is another example of a causative construction where only 'have' is possible. Can you work out why?