Yes, in preterm births and low/middle-income countries.
There is good evidence that early cord clamping leads to hypovolaemia, anaemia and low iron stores in the neonate. 
This seems to be more important when it comes to preterm births [2, 5]:
Delaying cord clamping was associated with fewer infants requiring transfusions for anaemia (seven trials, 392 infants; risk ratio (RR) 0.61, 95% confidence interval (CI) 0.46 to 0.81), less intraventricular haemorrhage (ultrasound diagnosis all grades) 10 trials, 539 infants (RR 0.59, 95% CI 0.41 to 0.85) and lower risk for necrotising enterocolitis (five trials, 241 infants, RR 0.62, 95% CI 0.43 to 0.90) compared with immediate clamping. However, the peak bilirubin concentration was higher for infants allocated to delayed cord clamping compared with immediate clamping (seven trials, 320 infants, mean difference 15.01 mmol/L, 95% CI 5.62 to 24.40) .
If mother has anaemia, delayed clamping should be considered:
Evidence from randomized controlled trials has shown that delayed cord-clamping is beneficial to infant iron status. [...] the association between the timing of cord-clamping and infant anaemia was modified by the mother's anaemia status. Significant benefits of delayed cord-clamping in preventing anaemia were found in infants born to anaemic mothers at both 4 months (aOR = 0.59, 95% CI 0.36-0.99) and 8 months (aOR = 0.38, 95% CI 0.19-0.76) of age .
A comment on the above quoted study gives some details on how delayed clamping contributes to baby blood volume:
Immediately after the birth of a baby, placental blood continues to flow in the direction of the child. The total fetoplacental blood volume is about 120 ml/kg of fetal weight, and the distribution of blood between fetus and placenta is roughly in a ratio of 2:1. This distribution remains unchanged if the cord is clamped early. Allowing placental transfusion to occur for at least 3 minutes results in greater infant blood volume (ratio 5:1). The rate of placental transfusion is influenced by the position of the delivered infant. From 10 cm above the level of the placenta (on the abdomen of the mother) to 10 cm below the level of the placenta (on the birthing bed), infants receive the maximum possible amount of blood for at least 3 minutes after birth. Keeping the infant 40 cm below the placenta hastens placental transfusion to near completion within 1 minute .
And it also claims the severity of child anaemia due to cord-clamping is exaggerated and there are other factors involved:
From a pathophysiological viewpoint, it is difficult to understand how the haematological benefits of delayed cord-clamping extend so far into infancy. Earlier work on cord-clamping has shown that by the age of approximately 6 months the infant outgrows its fetal iron reserves.4,7 From that age onwards, the infant becomes dependent on exogeneous iron sources, including iron-enriched infant formulae and weaning foods. Intercurrent infections, especially those of the gastro-intestinal tract, disturb iron absorption and might even increase iron loss from the gut .
WHO recommends delayed cord clamping in low and middle-income countries:
As per the WHO recommendation, delayed cord clamping in low and middle-income countries, may be beneficial [3,4], as many studies have documented improved iron stores during the first half of infancy, especially in resources-limited settings where iron deficiency anemia is highly prevalent. However, the benefits of delayed cord clamping in all term infants in industrialized countries need to be weighed against the possible need for more infants developing jaundice and needing phototherapy, especially in settings where “early discharge” is commonly practiced .
- Hutchon DJ. Immediate or early cord clamping vs delayed clamping. J Obstet Gynaecol. 2012 Nov;32(8):724-9. doi: 10.3109/01443615.2012.721030. PubMed PMID: 23075341.
- Rabe H, Diaz-Rossello JL, Duley L, Dowswell T. Effect of timing of umbilical cord clamping and other strategies to influence placental transfusion at preterm birth on maternal and infant outcomes. Cochrane Database Syst Rev. 2012 Aug 15;8:CD003248. doi: 10.1002/14651858.CD003248.pub3. PubMed PMID: 22895933.
- Blouin B, Penny ME, Maheu-Giroux M, Casapía M, Aguilar E, Silva H, Creed-Kanashiro HM, Joseph SA, Gagnon A, Rahme E, Gyorkos TW. Timing of umbilical cord-clamping and infant anaemia: the role of maternal anaemia. Paediatr Int Child Health. 2013 May;33(2):79-85. doi: 10.1179/2046905512Y.0000000036. PubMed PMID: 23925280.
- van Rheenen P. Less iron deficiency anaemia after delayed cord-clamping. Paediatr Int Child Health. 2013 May;33(2):57-8. doi: 10.1179/2046905513Y.0000000059. PubMed PMID: 23925277. Full text http://www.maneyonline.com/doi/full/10.1179/2046905513Y.0000000059
- Raju TN, Singhal N. Optimal timing for clamping the umbilical cord after birth. Clin Perinatol. 2012 Dec;39(4):889-900. doi: 10.1016/j.clp.2012.09.006. PubMed PMID: 23164185. Full text: http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3835342/