I was wondering whether there exists a study which measures the impact of economic news coverage to gdp or something like that
Nick Bloom, from Stanford, and some co-authors develop indices of economic policy uncertainty for the world's major economies. Their Economic Policy Uncertainty Index is based on newspaper coverage frequency of policy-related economic uncertainty. In case this is useful, their web page explains the methodlogy and the applications.
I have not heard of such a study and I have a feeling studying how a daily news item would affect an annual measurement value would be a pretty futile experiment.
However, there are some studies on how announcements from central banks or other financial authorities might affect expectations in the short run. My collaegue actually did her licensiate thesis on how ECB board announcements affected the growth path of euro inflation.
A pretty recent article would be Moessner, R.: "Effects of ECB balance sheet policy announcements on inflation expectations." in Applied Economics Letters and that might be a good place to start looking.