… could I run affiliate ads … in my podcast … and still ask the listeners in the beginning for contributions?
Yes, but this could be tricky to do well. It really depends on how you promote the affiliate. If you make the affiliate sound like an ad, some listeners will question why they need to donate if you already have an “advertiser.”
Listeners may also hope to stop hearing ads if they donate.
But if you include your affiliates in relevant contexts, they won't sound like ads to your audience, and thus gives you more freedom to ask for audience support.