In regards to your previous email, I have applied for a DBS check and received it this morning
No. Say eg In regard to your previous email, I applied for a DBS check and received it this morning. There is no need for Present perfect here.
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No.
Say eg In regard to your previous email, I applied for a DBS check and received it this morning.
There is no need for Present perfect here. You re just talking about two simple past actions.