This is my proof of concept but it works perfectly.
#r "nuget:Azure.Storage.Files.DataLake,12.0.0-preview.9"
using Microsoft.Spark.Extensions.Azure.Synapse.Analytics.Utils;
using Microsoft.Spark.Extensions.Azure.Synapse.Analytics.Notebook.MSSparkUtils;
using Azure.Storage.Files.DataLake;
using Azure.Storage.Files.DataLake.Models;
string blob_sas_token = Credentials.GetConnectionStringOrCreds('your linked service name here');
Uri uri = new Uri($"https://'your storage account name here'.blob.core.windows.net/'your container name here'{blob_sas_token}") ;
DataLakeServiceClient _serviceClient = new DataLakeServiceClient(uri);
DataLakeFileSystemClient fileClient = _serviceClient.GetFileSystemClient("'path to directory containing the file here'") ;
fileClient.DeleteFile("'file name here'") ;
The call to Credentials.GetConnectionStringOrCreds returns a signed SAS token that is ready for your code to attach to a storage resource uri.
You could of course use the DeleteFileAsync method if you so desire.
Hope this saves someone else a few hours of GoogleFu.