Good character

I’m a single parent with 2 kids 12 and 17 we have refugee settlement indefinite leave to remain from 09/10/2018 and we would like to apply for citizenship but I had a court penalty of misuse my mother’s blue badge on 28 /04/ 2018 , I don’t know if it will going to effect my situation as not good character or not ( I did DBS and it shows no police record ) , should I wait until it is over two years , also could I apply for my kids now and for myself later

Yes it would

Many thanks for your reply , so when should i apply to avoid the refusal

I don't know about you but for your kids, it depends with when the home office first became aware of your presence, when you first sought asylum or when you got it. You'd need to give your immigration history to be better advised.

also could I apply for my kids now and for myself later

Children under 18 and born abroad will only qualify when either parent does and the other parent holds ILR (if relevant). They cannot apply before you.

The 17 year old can wait till they are 18 and have held ILR for 12 months, done English B1 test and LUIK and apply on their own as an adult.

What you are telling doesnt have any impact on naturalization application.You shouldnt need to be scared and apply when due .Good character requirement is stopping at least 40%people to apply for naturalization from which HO is mostly targeting criminals &overstayers.If labour wins next election I think they will scrap this too harsh rule.