What B-lverk said pretty much.
If you're not using Man&Machine, the Canon rule is that if you remove Cyberware, you can not replace the lost essence with different cyberware. If you use Man&Machine, you -can-, but it is im-probable that you succeed. Here's a detailed example of how the rules work:
You decide to upgrade your Wired Reflexes from Basic-Grade rating 1 to Alpha-Grade rating 2.
First you remove the cyberware, which has a TN of 3 (this is the BEST time to try to regain the essence, due to a lower TN). You're in the run-of-the-mill type backstreet Black Market Clinic for a 0 point modifier. You have a little over 4 Essence worth of Cyber (including the Wires you're having removed) which adds a +1 to the TN. The conditions are mediocre, enough not to cause a modifier. So end result is a simple TN of 4. You assign 2 negative options and 1 positive (it's the only one that interests you) of: Fragility (you don't care how stressed the cyberware gets once gone) and Sensitive with the Positive Option of Essence Slot.
Doctor makes a BioTech skill (6) roll against a TN of 4 and rolls a 1, 2, 3, 4, 5, and 10. You got only 3 successes, so did not get the Positive Option.
Doctor continues surgery to Implant the Alpha Grade which has a Base TN of 4 and the same +1 because of low Essence, in addition to +1 for 2nd surgical procedure and another +1 for being Alphaware. TN 7 with 6 dice. Not likely (hence why you aim for Essence Slot at removal)....
Let's assume however that your Essence was 3 for the removal, or that he rolled 1, 3, 4, 5, 5, 10 giving you the 4 successes you needed for the positive option. Then you check the Threshold. Threshold here is NOT like Threshold of Manipulation spells. In surgery it simply means an increased TN.
Threshold for Essence Slot is +2 which is added to the Base TN of 4 giving you a TN of 6. As long as you get at least 4 successes at TN 4 and ONE of those 4 successes comes up 6 or higher, you got yourself an Essence Slot.
If you already had the cyber removed without using the Essence Slot option, then you're pretty screwed unless you get a top-notch scenario such as a team of doctors, a well stocked clinic/hospital etc (things most shadow runners don't have access to). Base TN is 4 to install, you need 4 successes to get a positive option, and the Doctor is probably rolling 6 dice (depending on how nice the GM is). If you can get the TN down to 3 you're in Luck and should opt for Essence Slot AND Essence Reduction. It doesn't change the TN for success, and if your Doctor rolls at least 1 7+ you can get That Alpha Wired-2 for 2.25 instead of 2.40 and fill in the removed cyber slot nicely (thus costing you a grand total of 0.25 essence).
Hope that wasn't too confusing, I just like explaining this kinda stuff.

Sphynx