Hi,
I have come across the following sentence:
"John came in a second time, flying about 500 feet overa farm near the village of Berg,..."
I believe "John came in the second time" is also possible.
If so, then what is the difference?
Both are possible, and sometimes no difference is intended.
However, with a second time the indefinite article stresses more that this is the first time this 'second time' has been mentioned by the writer.
You could say that it makes the statement more of a surprise to the reader.
eg He asked me to lend him $1000. I did.
Then he came back the next day and asked me a second time.
Generally speaking, 'for' is optional, but it depends on the particular context.
Best wishes, Clive