Translation:The girl needs permission to go into the city.
This is a tricky one. My judgement (and please look at my history and decide whether you want to listen to my judgement or not) is that the sentence is good.
Por implies purpose and the rule of thumb is that you need "por" before an infinitive when you can say "in order to" in English.
- Mi venis por vidi vin - I came to see you - I came in order to see you.
A second way to look at this is that it's similar to "for -ing" or "my true love for to see."
These rules of thumb break down in a very small number of cases, and there are one or two cases where one could quibble about whether "por" is really needed. If you learn the rule of thumb and the following pattern:
- Mi havas multon por fari
Then you will get more than 99% of the sentences right.
You'll see permeso por tio and permeso por X-i but they are not at all common compared to permeso X-i.
What is the difference between going to the city and going into the city?
Personally, I would use "to" when referring to the city like a point on the map ("I am going from London to Paris"), and use "into" when I am in the suburbs or outskirts of a city and making a trip into the city ("I am going into the city to see the play.").
"The girl needs a permission to go into the city" is not accepted.
Some say it that way, and it is a valid translation, thus it should be accepted.
Rather than posting this sort of thing all over the forums and cluttering it up, it would be way more constructive to report it using the "report a problem" button. In this case, I believe you're wrong. "She needs a permission" is not good English -- but if you're really concerned about the course, the best thing would be to use the button because the information goes directly to the course creators. They also receive notifications if some sentences are generating a large number of reports this way - so they can prioritize. Finally, they're busy reading reports and working on those sorts of things, so you don't often seem them reading these posts.