Undead Warlock's Spirit Projection states that both the body and the spirit is affected by damage and effects that apply to each equally. So from my understanding anything and everything that affects one affects the other, including equipment.
It does state that you don't replicate the equipment but the part that links the stats and effects would make me think that you can use your spellcasting focus and any equipment on you that has a passive ability.
"Your spirit can become untethered from your physical form. As an action, you can project your spirit from your body. The body you leave behind is unconscious and in a state of suspended animation.
Your spirit resembles your mortal form in almost every way, replicating your game statistics but not your possessions. Any damage or other effects that apply to your spirit or physical body affects the other. Your spirit can remain outside your body for up to 1 hour or until your concentration is broken (as if concentrating on a spell). When your projection ends, your spirit returns to your body or your body magically teleports to your spirit's space (your choice)."
My DM doesn't know exactly how to answer this with 100% certainty and I'm not sure either. Though I argue that the fact that it allows you to cast Form of Dread which allows you to frightened on hit and to use the Grave Touched to increase damage on hit in addition for the fact that it allows you to cast Conjuration and Necromancy without any S,V,or M's but it does not say anything about not having a focus.
Can I get some clarity about this? Thanks in advance.