broken EFI / missing space after re-partitioning — using mavericks 10.9.5 I have a few problems with a 2012 era macbook air. the first problem, and i'm not sure how to go about fixing this, is an issue i'm dealing with re: EFI.
When i select the hard drive in disk utility and attempt to verify it, I get the following error: 

"Problems were found with the partition map which might prevent booting
Error: This disk doesn’t contain an EFI system partition. If you want to start up your computer with this disk or include it in a RAID set, back up your data and partition this disk."

the second problem is trying to reclaim a missing 20GB of space that went missing when i deleted a failed Windows partition. The space just vanished, poof! You can see it in the diskutil list below.

 diskutil list


returns

       #:                       TYPE NAME                    SIZE       IDENTIFIER
       0:      GUID_partition_scheme                        *500.3 GB   disk0
       1:       Microsoft Basic Data EFI                     208.7 MB   disk0s1
       2:                  Apple_HFS MacOS                   479.0 GB   disk0s2
       3:                  Apple_HFS Recovery HD             650.0 MB   disk0s3


and running

gpt -r show /dev/disk0


shows

  start       size  index  contents
          0          1         PMBR
          1          1         Pri GPT header
          2         32         Pri GPT table
         34       2014         
       2048     407552      1  GPT part - EBD0A0A2-B9E5-4433-87C0-68B6B72699C7
     409600         40         
     409640  935546880      2  GPT part - 48465300-0000-11AA-AA11-00306543ECAC
  935956520    1269544      3  GPT part - 48465300-0000-11AA-AA11-00306543ECAC
  937226064   39878963         
  977105027         32         Sec GPT table
  977105059          1         Sec GPT header


EDIT:

I ran diskutil  resizevolume  disk0s2 limitson the two volumes (disk0s2 and disk0s3 respectively) and they returned as follows:

For device disk0s2 MacOS:
        Current size:  479.0 GB (479000002560 Bytes)
        Minimum size:  446.3 GB (446258642944 Bytes)
        Maximum size:  479.0 GB (479000002560 Bytes)

For device disk0s3 Recovery HD:
        Current size:  650.0 MB (650006528 Bytes)
        Minimum size:  650.0 MB (650006528 Bytes)
        Maximum size:  20.9 GB (20933816320 Bytes)


am i correct in reading this, that the missing 20 GB somehow ended up on the recovery HD?

so what's the first thing i should do? i want to keep my OS running on 10.9.5. I have a diskImage i downloaded from the internet, put onto a USB boot drive. I attempted to reinstall from this boot drive, to see if this would fix anything, but when i try to select the macOs, i get the error, "this disk cannot be used to start up your computer"

Besides this, the computer seems to operate fine, and has been working since april with this broken EFI and missing partition space.

any ideas?
THere's nothing actually wrong with your Air's EFI, what you are experiencing are the death-throws of a failing-and-fading-fast SSD module.

We tend to be rather blasé about solid-state drives these days, but seven years ago the technology was still maturing, and SSDs were still susceptible to sudden and inexplicable demise. When Jobs decided to start fitting SSDs into the first MacBook Airs, Apple did a lot of destructive data testing until the found a brand that they thought might last five years - Toshiba.

I'd look at getting a replacement SSD-stick - Transcend make a nice JetDrive replacement for your machine.