How do I make an action that mutates something and then opens another screen in react-admin?

Use redirect(), there is an example in the documentation: https://marmelab.com/react-admin/Actions.html#handling-side-effects-in-other-hooks

import * as React from "react";
import { useMutation, useNotify, useRedirect, Button } from 'react-admin';

const ApproveButton = ({ record }) => {
    const notify = useNotify();
    const redirect = useRedirect();
    const [approve, { loading }] = useMutation(
        {
            type: 'update',
            resource: 'comments',
            payload: { id: record.id, data: { isApproved: true } },
        },
        {
            onSuccess: ({ data }) => {
                redirect('/comments');
                notify('Comment approved');
            },
            onFailure: (error) => notify(`Comment approval error: ${error.message}`, 'warning'),
        }
    );
    return <Button label="Approve" onClick={approve} disabled={loading} />;