During austral summer, La Nina tends to produce cooler- and wetter-than-average conditions from around Madagascar to locations south and west. You can see these impacts on the climate.gov ENSO impacts page. This relationship tends to break down by around March. 

Regarding tropical cyclones, the relationship with La Nina is a bit challenging to discern. This study suggests that La Nina causes less frequent than normal cyclone occurrences, but this one also suggests that La Nina leads to stronger and longer-lived cyclones in the region. Therefore, the relationship between cyclones and La Nina over the southwest Indian Ocean may be complex and not fully understood.