It's not clear to me if the files are local (you're running the script on the server) or remote (on another machine). If local try running the command using a background job and pass in the credentials to Start-Job:
$job = Start-Job { Remove-Item -LiteralPath $path -force } -cred $cred
Wait-Job $job
Receive-Job $job
If they're remote, try using remoting:
Invoke-Command -computername servername `
-scriptblock { Remove-Item -LiteralPath $path -force } `
-Cred $cred
Note: This requires that you execute Enable-PSRemoting on the remote machine.
In general, putting raw passwords in your script isn't a great idea. You can store the password in an encrypted manner using DPAPI and later, only that user account can decrypt the password e.g.:
# Stick password into DPAPI storage once - accessible only by current user
Add-Type -assembly System.Security
$passwordBytes = [System.Text.Encoding]::Unicode.GetBytes("Open Sesame")
$entropy = [byte[]](1,2,3,4,5)
$encrytpedData = [System.Security.Cryptography.ProtectedData]::Protect( `
$passwordBytes, $entropy, 'CurrentUser')
$encrytpedData | Set-Content -enc byte .\password.bin
# Retrieve and decrypted password
$encrytpedData = Get-Content -enc byte .\password.bin
$unencrytpedData = [System.Security.Cryptography.ProtectedData]::Unprotect( `
$encrytpedData, $entropy, 'CurrentUser')
$password = [System.Text.Encoding]::Unicode.GetString($unencrytpedData)
$password