Need to find if particular hotfix is installed or not

I have a list of servers on which i need to find a particular KB ex: KBXXXX is installed or not. Output file should tell me if KB is installed or not.

Also, some of the servers will not be reachable as WMI might not be working or they are offline, so script should right error message as "WMI is not working" or "server is offline" in the output file

Can someone help me with a powershell script?

P SAsked:
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I'll give you an example script:

:: win8.1/server2012R2
ver |findstr /C:"6.3.9600" && reg query "HKLM\SOFTWARE\Microsoft\Windows\CurrentVersion\Component Based Servicing\ApplicabilityEvaluationCache\Package_for_KB4025333~31bf3856ad364e35~amd64~~" /v CurrentState |findstr 0x70 && goto end
:: server2012
ver | findstr /C:"6.2.9200" && reg query "HKLM\SOFTWARE\Microsoft\Windows\CurrentVersion\Component Based Servicing\ApplicabilityEvaluationCache\Package_for_KB4025343~31bf3856ad364e35~amd64~~" /v CurrentState |findstr 0x70 && goto end 
:: win7/server2008R2
ver | findstr /C:"6.1.7601" && wmic qfe get Hotfixid | findstr "4025341 4025337" && goto end
:: win10v1607/server2016
ver | findstr /C:"10.0.14393" && reg query "HKLM\software\microsoft\windows nt\currentversion" |findstr "0x5c8 0x5fc 0x639" && goto end
:: win10v1703
ver | findstr 15063 && reg query "HKLM\software\microsoft\windows nt\currentversion" |findstr "0x1e3 0x1f6 0x21c" && goto end
:: Server2008
ver | findstr /C:"6.0.6002" && wmic qfe get Hotfixid | findstr 4025409 && goto end
echo %date%>\\Someserver\logging$\!NotUp2date\%computername%.txt

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The strings in this script are for a particular patch - yours will be different and need to be changed.
P SAuthor Commented:
Thanks but I am looking for a powershell script for this.
And why would powershell serve you any better, if I may ask?
All the above commands work in powershell, too. It's just that && is not understood and would need to be exchanged for a powershell equivalent. But why care?

Let me tell you why I used batch: the powershell cmdlet get-hotfix did not work very well (didn't list updates as being installed, while in fact they were!), while the code above did. Don't use powershell for this.
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P SAuthor Commented:
I am new to powershell and trying to learn it. In my question, there could be 3 scenario's

1) WMI is working on the server but the patch is not installed

2) WMI is working but I don't have the access to query that server via WMI and in that case, I'll be getting "Access Denied" error

2) WMI is not working at all and I can't find the patch in that case

So i was wondering if someone could help me write a script as I am not sure if multiple condition can be applied in an "If" block.

I tried "Try" and "Catch" as well but it seems to be not working.

Example syntax:
Get-HotFix -id KB3192137 -computername remoteserver | out-file \\server\loggingshare\remoteserver.txt

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Output (if installed) would be
Source                Description      HotFixID             InstalledBy                           InstalledOn              
-------------                -----------             --------                  -----------                                  -----------              
remoteserver        Update           KB3192137     NT AUTHORITY\SYSTEM  9/12/2016 12:00:00 AM    

Output if not installed would be just an empty file (0 kB in size).

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