Strength of Hash obtained by XOR of parts of SHA3 I create a hash with SHA3-384. I split the resulting array into two. The first called head has the first 320 bits and the second called tail  has the last 64 bits.

Next, I XOR  head and tail. Which in fact means I XOR the last 64 bits of head with tail. This gives me a new hash of length 320.

The question is how is this resulting hash in terms of collision resistance compared to SHA-256 on one side and SHA-384 on the other side.

My goal is to produce a 320-bits hash. Is there a better way you would go about this?
Truncating a hash does reduce the collision resistance. Mixing in an independent string does not.

$$
\text{random thing} \oplus \text{independent thing} \to \text{random thing}\\
\text{random thing} \oplus \text{independent random thing} \to \text{random thing}
$$

Truncating hashes is common and the reduction is relative to the bits lost. The truncated $\text{sha3}_{384[320]}$ should be stronger than $\text{sha2}_{256}$ and is weaker than $\text{sha3}_{384}$.

I recommend that you do not try to mix in the truncated $64$-bits.