Unless the PC card and peripheral device both support the Power Delivery specifications, there is not much one can do on the card to achieve higher power delivery modes. And definitely not with the Type-A ports as in the referenced card. You can't just crank-up the voltage, you need an intelligent system that negotiates higher voltage and currents on both ends of USB 3.1 link partners.
What can be done however is to have (or design and build) an adapter that provides the variable power (from likely external source) in compliance to Power Delivery specifications. The adapter should include a dedicated circuitry that provides PD communication, and must convert the original Type-A to Type-C port (because the PD works only over CC lines, and VBUS protocol was abandoned). Obviously your peripheral device should understand this protocol and respond accordingly. However, I wasn't able to find this kind of devices on the market, only some HDMI adapters with side function of power delivery.
If your suffer a simple voltage drop along long cables and want to compensate it, the solution is either to use self-powered devices, or use a self-powered (via Ac-DC adapter) hub on the side of peripheral device (if the device can't be self powered).
As a last resort, if you have this PC card and can get it's schematics (or reverse engineer it), and determine that VBUS power supply (+5V) has a voltage regulator of adjustable type, you can beef-up the VBUS to 5.5V by changing corresponding resistor dividers. But without details of your particular PC card, no advice can be rendered.