Yes, there is a secondary qualification test, called the Navy Advanced Placement Test (NAPT), once known as the Nuclear Field Qualification Test (NFQT). The minimum requiring score is based on how you did on the ASVAB itself. When you take the ASVAB, it is comprised of 10 sections, 6 of which are used for creating your Nuclear Field qualifying score (and there are two different sets of 6 sections even). If you score over a certain benchmark then you will not need to take the NAPT. If you're under that requirement, but over the minimum requirement to take the NAPT then they'll have you take it and your minimum qualifying score will be determined by how far below you are. If you're below the miminum requirement to take the NAPT then they will not even offer the program to you at all.
As a rule of thumb I'd say that an ASVAB score of < 70 (not likely qualified), 70-85 (you'll need to take the NAPT), 85+ (probably won't need it). That is only a rough estimate though as your ASVAB score only accounts for 4 of the 10 sections (2 math & 2 english). I've seen an 88 who needed the test and a 71 who didn't. With a 81 on the EST (practice test) I wouldn't worry. Many of us who joined several years ago were required the take the test regardless of what we scored on the ASVAB and it isn't too difficult.