Putty works fine with Smart Card inside an RDP session.
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Putty application exe is excluded from Citrix hooking including the smart card hook.
The Putty executable is pageant.exe and is incorrectly included in the UviPRocessExcludes registry key.

As a result, no Citrix hooks including scardhook.dll (Smart Card virtual channel hooking) are loaded for the process. Without the Smart Card Hook there is no PIN prompt.
Key: HKLM\SYSTEM\CurrentControlSet\services\CtxUvi
Value Name: UviProcessExcludes
Type: REG_SZ
Value: badapp1.exe;badlongname.ex;badapp2.exe
See https://support.citrix.com/s/article/CTX107825-how-to-disable-citrix-api-hooks-on-a-perapplication-basis
Remove pageant.exe Entry from the UviPRocessExcludes key.
A reboot is needed for this to take effect.
Note – This scenario is not specific to Putty. If a specific application is working with Smart Cards, then the UviPRocessExcludes key should be verified.
One this is done it can be confirmed with Process Explorer that ScardHook.dll is loading in the process.

Smart Card PIN prompt should then be seen with the Putty Application.

The PIN prompt is seen when using the smart card with other applications inside the Desktop Session.
Issue is only with Putty