Mushrooms growing with my seedlings?!?

I have started seedlings in my south window of my home. They were started in the jiffy greenhouses, and I've moved them to newspaper pots. When I checked on the seedlings this morning, I found mushrooms growing in the flat! I know mushrooms are fungi, but how do I get rid of them? Are they going to bother the seedlings? My main concern is that they could be poisonous, I have a four year old and cat in the house. I've started seedlings this way for years, and have never come across this. The other weird thing is that they're only on the pots with the cosmos, not the zinnias, and they're in the same flat (don't know why that would make a difference?) Any advice you can give is appreciated.

(the long white strands in the picture are from my cat...she can't stay out of my plants!)

More mushrooms...

Hello there! I’ve seen your pictures! One thing I would like to ask: Are you using the pot-maker for those newspaper pots? It’s just a question non related to the mushrooms! I have one too and found out that the bad thing about it is that they get dry easily with the air and also absorb water from the soil they are containing, so be careful with that because it can get too compacted if you don’t keep it moist!
Anyway! What I see is that they are really humid (good environment for fungi) and you shouldn’t expect any problem with your seedling with this kind of mushroom. It would be good to open one of the pots and see what the mushroom is feeding on. What you see is the fruit so… it’s the last part of its life cycle. If you cut it down it should be enough to get rid of it (until the beginning of next cycle).
In this part of the cycle it needs light, air and humidity.
Now, I’m not an expert in mushrooms but I’ll try to find yours in my books to see if it is poisonous!
Please forgive me weird way of writing in English! I try to do my best!
I hope this information helps a little bit.
Take care.
Gus.

Hi, I don't have a pot-maker, I just wrap the paper around an old tupperware container I have, and fold them that way. They do dry out quickly. I think the lack of moving air may have something to do with the mushrooms, as well as the humidity, as you said. I've been picking the mushrooms off, and I'll be setting these outside in a couple of weeks anyway. Thanks for your help and all the info!
Rae Lynne