A sample script that shows how to find the Azure DevOps agent pool for a named agent. Also shows the call required to delete the agent

Get-AgentPoolForNamedAgent.ps1

function Get-AgentPoolForNamedAgent {
    param
    (
        $pat,
        $url,
        $agentName
    )

    $patToken = [System.Convert]::ToBase64String([System.Text.Encoding]::ASCII.GetBytes(":$($pat)"))
    $repoHeader = @{"Authorization"="Basic $patToken"}


    # get the pool ID from the name
    $uri = "$url/_apis/distributedtask/pools?api-version=7.1-preview.1"

    # usin the invoke-restmethod as it allow the same form to be used for get and delete calls
    $jsondata =  Invoke-RestMethod -Uri $uri -Method Get -ContentType "application/json; charset=utf-8; api-version=6.0" -Headers $repoHeader -MaximumRedirection 10

    foreach ($pool in $jsondata.value) 
    {

        Write-Host "Processing Agent Pool '$( $pool.name )'"
    
        # Get agents of an agent pool (Request method: Get):
        $pooluri = "$url/_apis/distributedtask/pools/$($pool.id)/agents?api-version=7.1-preview.1"
        $agentJsondata =  Invoke-RestMethod -Uri $pooluri -Method Get -ContentType "application/json; charset=utf-8; api-version=6.0" -Headers $repoHeader -MaximumRedirection 10

        foreach ($agentRec in $agentJsondata.value) {
            Write-Host "Processing Agent '$( $agentRec.name )'"

            if ($agentRec.name -match $agentName) {
                Write-Host "Agent '$($agentName)' [$($agentRec.id)] found in pool '$( $pool.name )' [$($pool.id)]"

                 # we are exiting the function here with the pool information but you have enough informaton to delete the agent using a call to
                 # DELETE https://dev.azure.com/{organization}/_apis/distributedtask/pools/{poolId}/agents/{agentId}?api-version=7.1-preview.1

                 # smethink like
                 # $deleteUri = "$url/_apis/distributedtask/pools/$($pool.id)/agents/$($agentRec.id)?api-version=7.1-preview.1"
                 # Invoke-RestMethod -Uri $deleteUri -Method Delete -ContentType "application/json; charset=utf-8; api-version=6.0" -Headers $repoHeader -MaximumRedirection 10

                 return $pool

            }

        }

    }
}

# Usage
# Get-AgentPoolForNamedAgent -url https://dev.azure.com/yourorg -agentName my-agent -pat <yourpat>