Tricky, but may be possible, provided your web hosting provider allows you to change DNS records, specifically the MX record. This is still a kind of messy procedure, and will probably not provide desirable results.
You have to keep your primary DNS at the web-hosting provider (that is, your domain registration nameservers should point to your web hosting provider). The reason for that is that DH doesn’t allow you to futz with the web hosting portion of your domain record.
Once you’ve added the domain to DH you’ll be able to see its associated DNS entries in the DNS section of the DH control panel; copy the the “A” record for mail (an IP address) and the MX records (usually two FQDNs) into the DNS control panel of your web hosting provider.
Caveat 1: DH probably won’t be able to offer much in the way of support for such an arrangement.
Caveat 2: Popular ISPs like AOL, Comcast, etc. will probably NOT accept mail from you as this arrangement will almost certainly foul the rDNS (reverse DNS) lookup of the record which they require to match your primary domain record - this is a restrictive anti-spam measure that I’ve run into before.