I am having trouble at the moment with my new mincer because bits of connective tissue keep getting caught in the works and clogging it up, stopping the mince coming out. However I'm not quite sure how to prevent it. I try to take obvious bits off, but a lot of it is in very thin sheets which are challenging to remove, especially when the meat is cubed.
My husband and I regularly grind our own meat. What we do is trim the meat before cubing. If any connective tissue / silver-skin is on the outside of the meat or running through the meat we are getting it off of a larger piece of meat, which is much easier because you can get a better hold on it.
We also use a filet knife for trimming the tough tissue. It allows you to trim much closer and with much more control, minimizing any waste.
And last but not least, chilling the meat helps also. While you don't want it frozen, firm is easier to handle.