If all you want to report is your NIC details, triggering on the presence of the software but no need to display details, just add a check to the WHERE clause to see if the software match exists.
...
Where
tblNetwork.Description Like '%OUR SPECIFIC NETWORK ADAPTER%'
And tblAssetCustom.State = 1
And tblComputersystem.Domainrole < 2
And Exists (SELECT tblSoftware.AssetID
FROM
tblSoftware
INNER JOIN tblSoftwareUni ON tblSoftwareUni.SoftID = tblSoftware.SoftID
WHERE
tblSoftware.AssetID = tblAssets.AssetID
AND tblSoftwareUni.softwareName Like '%OUR SOFTWARE TITLE%')
...
If you
do want to see software details, you'll need to JOIN that.
...
tblNetwork.DNSHostname,
tblNetwork.IPSubnet,
OurSoftware.SoftwareName,
OurSoftware.SoftwareVersion
From
tblAssets
Inner Join tblAssetCustom On tblAssets.AssetID = tblAssetCustom.AssetID
Inner Join tsysAssetTypes On tsysAssetTypes.AssetType = tblAssets.Assettype
Inner Join tblComputersystem On tblAssets.AssetID = tblComputersystem.AssetID
Inner Join tblNetwork On tblAssets.AssetID = tblNetwork.AssetID
INNER JOIN (SELECT
tblSoftware.AssetID,
tblSoftwareUni.SoftwarePublisher,
tblSoftwareUni.SoftwareName,
tblSoftware.softwareVersion
FROM
tblSoftware
INNER JOIN tblSoftwareUni ON tblSoftwareUni.SoftID = tblSoftware.SoftID
WHERE
tblSoftwareUni.SoftwareName LIKE '%OUR SOFTWARE TITLE%') AS OurSoftware ON OurSoftware.AssetID = tblAssets.AssetID
Where
tblNetwork.Description Like '%OUR SPECIFIC NETWORK ADAPTER%'
And tblAssetCustom.State = 1
And tblComputersystem.Domainrole < 2
...
The downside to including the software details is that if you're not precise enough in your filtering, you could end up with multiple rows of output.