The SMPS500R is the power supply module.
"half bridge" is a term I've never seen before. It means 2 channel (stereo). I'd also call it unbridged.
Bridged means that the amps are driven by the same signal, but inverted in phase, so when the speaker is connected across the 2 amps, it sees up to 2X the effective voltage swing, for up to 4X the power. (You never get 4X the power in reality.) https://en.m.wikipedia.org/wiki/Brid...led_amplifiers
This Wikpedia link isn't the clearest for a newcomer, but it contains the details.
The reference design (https://www.infineon.com/dgdl/irauda...53569addc92bfb
) says that you can set it to operate with an 8 ohm speaker in bridged mode using the switch S300. I don't see a reference to the speaker output connections. If you have access to a simple DVM (digital volt meter), check whether one of the output terminals is connected to ground. Say, the negative (-) terminal. Change S300 to the bridged position, and connect the speaker across the terminal that are not grounded. Or you could just try it. Connect between the two plus terminals, or the two minus ones. If you choose the wrong pair, nothing bad will happen, just no sound until you switch to the other pair.
Don't bridge with a 4 ohm speaker. Bridging causes the amplifier to see half the speaker impedance, and the amp may not work with a 2 ohm load.
If you have a friend who is more familiar with audio, it may be a good idea to get a second pair of eyes. You don't want to damage a €400 speaker.