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For anyone else who comes across the same problem, this worked for me:

// include the library code:
#include <SPI.h>

// Set Constants
const int adcChipSelectPin = 8;      // set pin 8 as the chip select for the ADC:

// Start setup function:
void setup() {
    pinMode (adcChipSelectPin, OUTPUT);
    
    // set the ChipSelectPins high initially:
    digitalWrite(adcChipSelectPin, HIGH);

    // initialise SPI:
    SPI.begin();
    SPI.setBitOrder(MSBFIRST);         // Not strictly needed but just to be sure.
    SPI.setDataMode(SPI_MODE0);        // Not strictly needed but just to be sure.
    Serial.begin(9600);

    //Clock Speed: Master clock/divider
    //84Mhz/64 =  1.3 MHz
    SPI.setClockDivider(SPI_CLOCK_DIV64);

    //Send dummy values over when ADC is turned on 
    //Hold Din High for 16 clock cycles twice 
    digitalWrite(adcChipSelectPin, LOW);
    SPI.transfer(0xFF);
    SPI.transfer(0xFF);
    digitalWrite(adcChipSelectPin, HIGH);

    digitalWrite(adcChipSelectPin, LOW);
    SPI.transfer(0xFF);
    SPI.transfer(0xFF);
    digitalWrite(adcChipSelectPin, HIGH);
    asm volatile("nop"); 
    asm volatile("nop"); 
    asm volatile("nop"); 
    asm volatile("nop"); 
    asm volatile("nop"); 
    //Send dummy values, next read should have correct values

    digitalWrite(adcChipSelectPin, LOW);
    byte FirstByte = 0x83;
    byte SecondByte = 0x50;
    byte response1 = SPI.transfer(FirstByte); 
    byte response2 = SPI.transfer(SecondByte); 

    digitalWrite(adcChipSelectPin, HIGH);

    asm volatile("nop"); 
    asm volatile("nop"); 
    asm volatile("nop"); 
    asm volatile("nop"); 
    asm volatile("nop"); 
    ///////////////////////////////////////////////////////////////////////////////////

    //50ns quiet time?
    asm volatile("nop"); 
    asm volatile("nop"); 
    asm volatile("nop"); 
    asm volatile("nop"); 
    asm volatile("nop"); 
} // End setup function.

// Start loop function:
void loop() {
    float voltage[2];
    for (int i = 1; i < 17; i++) {
        voltage[i] = readAdc(0, 1);
        //50ns quiet time? 
        asm volatile("nop"); 
        asm volatile("nop"); 
        asm volatile("nop"); 
        asm volatile("nop"); 
        asm volatile("nop"); 
    }
} // End of loop function.

//Function to read the ADC, accepts the channel to be read.
float readAdc(int channel, int slave) {
    float value;
    if (slave == 0) { 
        byte FirstByte = 0x83;
        byte SecondByte = 0x50;
        FirstByte |= channel << 2;
        
        noInterrupts(); // disable interupts to prepare to send address data to the ADC.
        
        digitalWrite(adcChipSelectPin, LOW); // take the Chip Select pin low to select the ADC.
        //Write = 1
        //SEQ = 0
        // ADD3-ADD0 = 0000
        //PM1 = PM0 = 1
        //Shadow = 0
        //WEAK/Tri = 1
        //Range = 0
        //Coding = 1
        //0b10000010 1010 0000
        //extra 4 zeros to fill

        //May need to wait 16 * (1/freq)
        //16 * (1/1.3MHz) 
        byte response1 = SPI.transfer(FirstByte); 
        byte response2 = SPI.transfer(SecondByte); 
        digitalWrite(adcChipSelectPin, HIGH);     

        interrupts(); // Enable interupts.
        
        //Serial.println(response1);
        Serial.println(response2);
        byte channel = (response1 >> 4);
        //byte value = (response1 << 4);
        //byte digitalValue = (response1 << 4) | response2; 
        //Serial.println(digitalValue);
        //Serial.println(channel);
        
        byte digitalValue = 0x00;
        float value = (float(digitalValue) * 3.3) / 4096.000; // The digital value is 
        converted to an analogue voltage using a VREF of 2.048V.
        //Serial.print(channel);
        //Serial.print("\t");
        //Serial.println(value);
    }
}