Have you checked the admissions criteria for that particular school? Some schools state that children currently attending x infant school get priority over children attending y, even if they are not linked.
This isn't always the case though.
I would imagine that most parents in your area would be applying for the Junior School nearest to or linked to their infant school and they are likely to have the same amount of places in their intake so it is likely she would get a place.
It wouldn't harm to put a covering letter in. They can take into account the specific needs of your child (emotional/health etc) when allocating places.
Definitely put in a covering letter. I wouldn't lie about your address as they can check it easily if they are in doubt (and presumably her current teacher knows your circumstances have changed) but do mention the upheavals she has recently gone through, situation with ds at adjacent school etc. I know someone who used the fact that their CM already collected other kids at a particular school as an argument to get in. Is the school oversubscribed usually ?