For present tense, it's really easy: There is no present tense for perfective aspects of Russian verbs. мыть is imperfective, помыть is perfective, so you can only use мыть here. You literally cannot use помыть to translate this sentence.
As explained in
Imperfective - Incomplete, ongoing, habitual, reversed or repeated actions
Perfective - Actions completed successfully.
When you're talking about present tense, activities have to be on-going. It doesn't make sense to talk about something going on at the present moment which has been completed and thus is not going on at the present moment.
По is a prefix. This time, помыть (or вымыть) is a perfective counterpart of мыть. It means a single action finished at some point in time. You use these to express finished actions in the past or the future and also to tell stories in a sequence (i.e., when one action follows another).
Мыть, on the other hand, expresses a range of more blurry meanings like ongoing, repeated or habitual action, the name of the action or action mentioned without focusing on when exactly it happened.
English does not have this perfective/imperfective distinction in its aspectual system, so understanding the basics of it will take you some time.
Well, нам надо помыть = we need to wash. In this case "to wash" is a whole, complete, singular action. This thing (wash) needs to be DONE. Speaking of it being done, ie completed, puts it in perfective.
If you were to say something like, we need to be washing, that would be нам надо мыть as far as I understand. As "washing" is an ongoing action, imperfective.
Thank you Jeffrey, that website really helped understand that. But in the website they give an example that I don't understand and would really appreciate if you could explain it to me. They say that "Говорить" is imperfective and its perfective is "Сказать". But, from this course, I got the idea that "Говорить" was "speak (a language)" or "talk" and that "Сказать" was "tell" or "say". I would be really thankful if you could clear my doubts.
The position of 'now' in the English translation has got me thinking.
It's not incorrect, but there's something about it that just doesn't feel right.
After some thought, I've decided that it might be because 'now' in that position seems very formal (and probably more likely to be written than spoken), but 'kids' is very informal, so it could be this mismatch of register that is the problem.
'The children/kids are washing spoons now' and 'Now, the children/kids are washing spoons' sound much more natural to me.
That said, the way in which this is said can convey quite different emotions, from simple, factual comment on the children's current activity, to exasperation/amusement at what is just the latest in a series of naughty/mischievous/amusing forms of behaviour. I don't know if the same is true of the Russian sentence or not.