This is more of a POC on adding or editing ACLs via PowerShell. This will add the account 'NT AUTHORITY\System' to have FullControl access to the folder in question. This also is looking for a specific UNC, but that can easily be edited.

Add-SystemToFolderACL.ps1

<#
.Synopsis
   This Function will add "NT Authority\SYSTEM" to a folder ACL.
.DESCRIPTION
   This Function will add "NT Authority\SYSTEM" to a folder ACL, specifically to \\folder\path\.
.EXAMPLE
   Add-SystemToFolderACL \\folder\path\user1
.EXAMPLE
   Add-SystemToFolderACL user1,user2
#>
function Add-SystemToFolderACL
{
    [CmdletBinding()]
    Param
    (
        # Param1 help description
        [Parameter(Mandatory=$true,
                   Position=0)]
        [Alias('Folder')]
        [string[]]$Path
    )

    Begin
    {
        # settings to allow full control
        $group = 'NT AUTHORITY\SYSTEM'
        $rights = [System.Security.AccessControl.FileSystemRights]"FullControl"
        $access = [System.Security.AccessControl.AccessControlType]::Allow
        $inherit = [System.Security.AccessControl.InheritanceFlags]"ContainerInherit,ObjectInherit"
        $propagate = [System.Security.AccessControl.PropagationFlags]::None
        $ace = New-Object System.Security.AccessControl.FileSystemAccessRule($group,$rights,$inherit,$propagate,$access)
    }
    Process
    {
        foreach ($p in $Path)
        {
            # Break if NT AUTHORITY\SYSTEM is already present
            if ($p -like '\\folder\path\*' -and (Test-Path $p))
            {
                if (((get-acl $p).Access).IdentityReference -contains 'NT AUTHORITY\SYSTEM') { break }
                $acl = Get-Acl $p
                $acl.AddAccessRule($ace)
                $acl.SetAccessRuleProtection($false,$false) # preserves inheritance ($true,$false) will disable inheritance
                Set-Acl $p $acl
            }
            elseif (Test-Path "\\folder\path\$p")
            {
                if (((get-acl "\\folder\path\$p").Access).IdentityReference -contains 'NT AUTHORITY\SYSTEM') { break }
                $p = "\\folder\path\$p"
                $acl = Get-Acl $p
                $acl.AddAccessRule($ace)
                $acl.SetAccessRuleProtection($false,$false) # preserves inheritance ($true,$false) will disable inheritance
                Set-Acl $p $acl
            }
            else
            {
                Write-Warning -Message "Path not found! Verify that the folder exists and is spelled correctly."
            }
        }
    }
    End{}
}