T7810 PCI-E X16 output power

PCIe socket standard power limit is 75W (for any of x1 x4 x8 or x16 power is the same - supplied by first segment of connector).
The 225 W limit is via additional power connector directly into PCIe device (like with GPUs) and is imposed because of PSU constraints for T7810 as MB without CPU but with all drives and RAM (PCH, other controllers incl. USB, Network, PWM/VR etc.) takes at least 150W, then CPU in turbo-boost + AVX512 mode can take close to 250W leaving very limited spare power capacity for PCIe devices from default power supply unit of 685W (or 825W units with advanced configs, or even 1.3KW as in your other forum thread).

T7810 are quite old units and over the time (especially if used under heavy stress workloads) characteristics of electronic components (and conductors/connectors) may deteriorate leading to failures or abnormal /unstable operations (e.g. weakening connections/fractures in soldering, etc.) , this is expected behavior and eventual outcome.
It's normal practice to decommission old units and replace with new ones (with warranty).
Probably the only alternative is to consider improving airflow to adequate levels over the overheating surfaces, but if power lines connection is unstable {e.g. oxidized contacts, sparks, etc.} then it will only get worse after some time very often leading to off-warranty failures on connected devices.

The direct output power from PCIe slots of your system board are:

Slot 1, 3, and 5 provide maximum 25w each

Slot 2 and 4 provide maximum 75w each

These info are imprinted on the board itself.  The numbering in order with slot 1 is the closest to the CPU.  Slot 2 and slot 4 are primarily intended for graphics card uses and can take additional power from PCIe cables (also known as PEG) to provide up to 225w on each slot.  All slots are supporting low power cards such of NIC, HBA.

Slot 2 is the first choice for graphics card.  So move and test your network card on slot 4 which has better air flow from system fan.  Also, your tesla P4 was designed to use with massive blow through air cooling from server rack and slot 2 of T7810 does not have adequate air movement to cool it off.  It can not be related and comparing with the overheat NIC card, and certainly not causing by the PCIe slot.