1. Let's first look into the relation of lipid biosynthesis and peroxisomes -
According to this article, the main functions of peroxisomes are-

As you can see most of the pathways that occur are oxidative in nature. That is, the main function is breaking down lipids and not their synthesis. The reason why peroxisomes are also known as Oxidative organelle. Even though peroxisome is involved in the synthesis of plasmalogens, and it has no requirement of hydrogen peroxide for the same.
Endoplasmic Reticulum is the organelle that synthesizes majorly lipids. So, the deficiency of hydrogen peroxide won't affect lipid synthesis.
2. Next is the role in the pentose phosphate pathway-
The major pathway for glucose catabolism is glycolysis. The location of the glycolytic pathway is the cytoplasm of the cell. Parallelly to glycolysis, PPP also runs in the cytoplasm. However, the major role of PPP is anabolic and not catabolic. The PPP is generally studied in 2 phases -
- The oxidative Phase (Generation of NADPH when Glucose is converted to 6-phosphogluconate)
- The non-oxidative phase (Synthesis of 5-carbon sugars)
As this article states
The glycolysis and pentose phosphate pathways (PPP) are tightly connected. The glucose entering the in cell membrane is rapidly phosphorylated by HK and converted to G-6-P. The G-6-P is metabolized either by the glycolytic pathway, generating pyruvate and lactate, or by PPP to produce NADPH. The F-6-P and G-3-P generated from the non-oxidative branch of PPP can enter glycolysis
Also,
The R-5-P, a product of oxidative PPP, is converted into G-3-P and F-6-P by transketolase (TKT) and transaldolase (TALDO1), creating a reversible metabolic link between PPP and glycolysis
Since, both of these metabolic pathways are highly connected to each other. The major location for PPP is the cytoplasm. However, PPP also do occurs in plastids and peroxisomes.
Again, if there is a deficiency of hydrogen peroxide in peroxisome it won't affect the PPP.
- Role in the breakdown of fatty acids
As this article states -
Oxidation of fatty acids occurs in multiple regions of the cell within the human body; the mitochondria, in which only Beta-oxidation occurs; the peroxisome, where alpha- and beta-oxidation occur; and omega-oxidation, which occurs in the endoplasmic reticulum.
Also, as mentioned above peroxisomes are the main location for fatty acid oxidation.
Let's take a look at how hydrogen peroxide has a role in this catabolism.
From Lehninger's Principles of Biochemistry(7th edition) -

In the very first oxidative step (for mitochondria), the H-electron is transferred to oxygen, and water is produced.
In the case of the first oxidative step (for peroxisome), the H-electrons are first transferred to FAD, making it FADH2 then to oxygen, generating a hydrogen peroxide molecule. This biochemical reaction is catalyzed by H2O2 producing enzymes. This H2O2 is further broken into water and 1/2Oxygen by catalase, thus canceling out the reactivity chances of H2O2 with any other molecules.
If there is a deficiency of H2O2 in peroxisomes, that means any of the enzymes catalyzing the formation of H2O2 is deficient. In such a scenario, the first oxidative step of fatty acids breakdown won't take place in peroxisomes. This will eventually lead to the accumulation of long fatty acids chains in peroxisomes and further apoptosis.
Mitochondria's main role is in energy production and not fatty acid oxidation, though it will help in fatty acid oxidation but the number of fatty acids oxidized only by mitochondria will be very very less and the net result will be an accumulation of fatty acids.