The first thing to do is look around the trailer park to see what other people have done on similar trailers. Also check with the trailer park management to see what they allow. As long as you are consistent with other patio roofs in the area, you should be OK.
Find out if you need a building permit from the city to build this. Many communities require a permit for things like this, even fences which many in the community may not be aware of.
It looks like you will need a 12' span so 2x4's will not be adequate, even for a deck roof.
I would also use 4x4's for posts, one every 6' so you will need 10 of them. Make 5 of them, the ones on the outside, about 2.5" shorter than the inside ones. The finished height of the roof should be a little lower than the rain (drip) edge of the roof of the trailer. The rain edge is a little U channel at the base of the roof, most trailers of this type had them, or had a section of U channel above the doors anyway. If it is the type with the U channel at the top of the door, then the rood will have to be taller.
Next take an inner post and an outer post and connect them together with a 12' 2x6. Cut the 2x6 about 6" short so that it is 11.5' long. This will be flush with the edge of the inside post but stick out over the outside post. The posts must fit on the deck so they should be spaced about 9.5' apart. Build 5 of these and stand each one up as it is built and hold in place with a temp brace.
Use 2 12' 2x6's or one 24' 2x6 at the outside edges of the 2x6 rafters to make a fascia. Now you can lay 2x4's flat on top of the rafters every 16 to 24". Since the span will only be 6', that should not sag. You will want the 2x4's to stick out over the end rafters about 16".
Before you remove the temporary bracing, you need to make corner braces. Use 2' length of 2x4 with the ends cut at 45° and use them at each corner of the roof between the posts and the rafters and between the posts and the flat 2x4's just above the posts. You will need a minimum of 8 braces, but putting more braces at the center posts and rafters/roof deck boards would not hurt. Then remove the temp bracing and toenail the 4x4 posts to the deck.
Now lay the choice of roofing, metal or fiberglass, across the roof deck boards. Screw them down to the roof deck boards, there are special screws with rubber seals for this.
Note, this roof is very minimal as far as loading goes. If you have any significant snow load, everything will have to be beefed up. You may need 7 ribs (a rib is two posts and a rafter) spaced every 4' and you may have to use 2x8's in place of the 2x6's. You may even have to use plywood over the deck boards before laying the metal, or space the roof deck boards on much closer spacing, 6 to 12" instead of 16-24". It all depends on how much snow you could expect to get where you live.
The last thing, since this is a free standing roof, you may want to put a bridge between the patio roof and the trailer roof over the door for rainy days. If the trailer roof has a drip channel at the base of the roof where you can put a piece of sheet metal under the drip edge but on top of the patio roof, that will help. It needs to extend at least 6" each side of the door. Aluminum flashing should do for this. Be sure that the attachment to the trailer does not cause a leak in the trailer.
Note, this is just one way to do this. Look around the trailer park to see how others have done this but be aware that many older patio roofs will be attached to the roofs of the trailers because that is how it used to be done, New research has shown that this is not a good practice.
Thank you so much. This may actually help some more. Will check around and continue to do some more research on this project. :)
One more thing, with this or similar type roof, you will be able to add a railing system either outboard of the posts or between them. At 31" high, most codes do not require a railing system, but 31" is a long way to fall, especially for the aged.
With this type of trailer there won't be enough headroom to combine with an adequate roof pitch unless the deck is at least 2 steps down from the trailer floor (~14-16'). What I've done in these situations is to frame and cover a short vertical "knee wall" above the edge of the trailer just high enough to give me the needed roof pitch while maintaining headroom at the outer end. Note that I do this with a free-standing structure- there is no connection to the trailer so there will be a gap for rain to drip down through. Most places will not allow you to attach a deck, stairs, or exterior addition of any kind to a trailer. To keep the rainwater at bay, about an inch above the trailer roof I attach an "L" shaped flashing to the knee-wall. Now the water will only run down the wall, not drip. If you want to bug-proof that gap make it a bit tighter and roll up some screening to stuff in the gap. That's legal because it's not attached to the trailer but it will stay in place and will allow water to pass through. Even if you can, I'd recommend against sealing this area into the trailer because it will retain water and you're going to have problems down the road at that joint- metal trailer roofs are not meant to hold water, only to allow it to run off. The span is definitely too long for single 2X4 rafters but if you sister two together then set them 16"OC they would do fine- this may not be code-worthy where you are so check on that.
Based on the stock of material you have, a better option might be to do a "Roof-over" above the entire trailer. Looks great, eliminates possible roof leaks, and keeps the sun from heating the trailer roof in the summer so AC bills are much lower. Then the roof on the deck side could be extended straight from the center ridge giving you the needed headroom height automatically. Roof-overs are a big thing here in upstate SC with our scorcher summers and with the gable ends closed in the trailer looks almost like a house. It's the biggest bang-for-your-buck you can get with a trailer!