First you will have to decide whether it is the right person. Pimlico is close to, but not in, Fulham; 'Fred John E' is not the same as 'Frederick James'; and someone born in that quarter would have been 8 in 1911 not 9. However, I don't think there are any other strong candidates for Fred John E Eveleigh on the 1911 census.
Even if it is the right person the certificate will almost definitely not name the father (and, since they didn't marry until 1906, Joseph is unlikely to be the father anyway). What it might do is confirm that Fred is the illegitimate son of Emma. I'm not an expert on Bastardy Recognizances, but I think 1906 is well past the date when they existed, and the magistrate records have probably been destroyed, so there is no way back there either.
Richard