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Yes, you will need a fungicide.

With wet weather comes many fungal infections. This tree is usually tolerant of disease unless it is too young. What can be done is to apply a HANDFUL of DOLOMITIC LIME and a TEASPOON of WETTABLE SULFUR to the soil around the tree. This can be done once or twice per year, if you know the season might be wetter than usual.

You will also want to spray the top with a fungicide. This article will help: https://www.gardeningknowhow.com/garden-how-to/info/using-fungicides-in-garden.htm

You willow appears to have been grafted. The original top has died off, so you will not have the tree that you bought. It will be a different variety of Willow that was used for the root stock.

This article will help you to care for willows: https://www.gardeningknowhow.com/ornamental/trees/willow/grow-a-willow-tree.htm

Yellowing does indicate watering issues; either too much or too little.
Watering. In general, a newly planted weeping willow requires 10 gallons of water applied two to three times weekly for each inch of the trunk's diameter. After the first month, you can reduce watering to once a week
https://www.gardeningknowhow.com/ornamental/trees/willow/grow-a-willow-tree.htm

This is a rust fungus! They can be easy to get rid of, though. Dispose of what you can, then treat with a fungicide.

These articles will help:

https://www.gardeningknowhow.com/plant-problems/disease/learn-about-plant-rust-disease-and-rust-treatment.htm

https://www.gardeningknowhow.com/garden-how-to/info/using-fungicides-in-garden.htm

Every trimming will introduce a new infection if done at the wrong time, and with anything other than clean sterile shears.

It is likely that it is being pruned out of season, or with unsanitary implements. Make sure that you are pruning, either, in early Spring or during dormancy to avoid introducing infections.

This article will offer more information: https://www.gardeningknowhow.com/ornamental/trees/willow/how-to-prune-willow-trees.htm