I'm wanting to change out our old faucet (second photo here) which has a connection underneath for the sink sprayer between the hot and cold supply (as shown in the first photo). I messed up when I ordered the new faucet, which doesn't have such a connection, but am hoping for any advice/suggestion on what I might be able to do in order to plumb in the new faucet with a sink sprayer that can supply hot or cold water like the old one. Last photo is the new faucet.

you goofed

return the 'new' faucet and get a three hole model with a spray

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"Measure Twice & Cut Once" - Retired U.A. Local 1 & 638

You can't adapt the new faucet to add a side sprayer. The faucet would need an internal diverter valve as well as a connection port for the hose. A stand alone sprayer faucet, like used in a hair salon, would work. Your existing faucet is using 3 sink holes. Are going to just install a plug in the center hole? The new faucet is 8" hole spacing?

best bet is to return the faucet you ordered and get an integrated pullout or pulldown faucet as steve suggested. The old hand sprayer hole can be covered or repurposed to use a soap dispenser, or a dishwasher air gap.