A few things about your numbers...
GDP per capita is not average income. Median net income in the EU was about $18 650 in 2017.
The EU measures poverty on a relative basis (see o.m.'s answer). Specifically it defines it as 60% of the median income. So around $11k (close to your figure).
A poverty rate on this basis is about 17% as you say. Specifically, this means that about 17% live on less that 60% of the median income.
As a consequence of the above, your calculations don't really make much sense as poverty does not include GDP in its measurement.
However, one could work out roughly what's needed. Assume that the mean poverty income is $8k. There are ~750mm people in the EU and the GDP is about $18.8tn. On that basis it would cost ~7% of GDP to move everyone below the poverty line up to the median income.
Would this eliminate poverty? Technically yes as there would now not be anyone below the poverty line and you're not moving the median income appreciably.
Would this be desirable? Highly questionable.
Firstly, any significant dent in the health or education of the workforce would also dent GDP so the relative percentage needed would increase.
Secondly, it becomes uneconomic to earn less than the median salary. You're better off earning poverty wages (or nothing) and get the government top up. So it wouldn't be 17% you'd need to find the money for but a much larger percentage.
Thirdly, there are much easier ways. The traditional left wing way is to substantially increase taxation for the middle classes. This depresses the median net income so the poverty level reduces. You then have a lot less people to find money for to reduce poverty levels. Obviously, it doesn't actually make anyone's life better but it looks like it does.