Does the Sona-TI351 have to be supplied with a 32kHz clock signal?

Yes, the Sona-TI351 requires a 32.768 kHz signal to function.

That signal can be generated internally by the chip-set itself by leaving the SUSCLK / SLOW_CLK_IN pin unconnected. This comes with the benefit of having less components on the PCB. But also comes with the caveat of being less accurate and more power consuming compared to sourcing that signal externally.

For optimal power consumption, the slow clock can be generated externally by an oscillator or sourced from
elsewhere in the system, e.g. a GPIO from the host processor.

The external source must meet the requirements shown in below table and must be be fed into the SUSCLK / SLOW_CLK_IN pin and must be stable before nReset is de-asserted and device is enabled.

External Slow Clock Requirements