I think your question would be better worded as, "can I get an infection after receiving the flu shot?" Since a secondary infection would imply that you had an infection in the first place.
Your question is not uncommon, and many people hesitate to receive it again after experiencing what you are describing.
Some patients do report very mild flu-like symptoms after receiving the flu vaccine via a shot, but the symptoms are much more common with the nasal spray. Symptoms may include feeling sleepy, light cough from throat irritation, achy, headache, tired, mild fever, joint point. However, these symptoms are not due to infection, the symptoms are from your immune system reacting to the dead virus particles (eww) to the vaccine. I'm not making this up, it's called the inflammatory response. The inflammatory response symptoms are GOOD, though, because it means your immune system is doing its job like it is supposed to.
Once your immune system has been in contact with these dead virus particles it will have learned how to combat those strains of viruses in the future. Your system begins to make what are called antibodies, which are what protect you in case you do come in contact with a live virus. In fact, these antibodies are called ig-M, the M actually stands for memory, neat huh? Your white blood cells will actually remember those strains of virus and easily destroy them on contact. So when and if you actually come in contact with the flu, your white blood cells will be ready to do their job very quickly.
It is certainly possible to get sick during any period of time before or after receiving the flu vaccine. Your body takes several weeks to build immunity to the flu virus after receiving the flu vaccine. It is still possible to contract the flu during that period. In fact, it is possible to contract the flu even after your body achieves immunity, because the flu virus changes constantly even within 1 flu season. The advantage to the vaccine is that if you do happen to get sick, the flu will not be as severe since you will have extra antibodies to begin with.
The bottom line is that it is your body and you know it best. If you have concerns I would certainly recommend seeing a physician.